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H1 Math

H2 Math

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N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>An adventure park is designing a new zipline that will span between two hilltops. The diagram below shows a vertical cross-section of the zipline setup. Points are defined relative to a main control centre at $(0, 0, 0)$. The ground is horizontal and is modelled as the $x$-$y$ plane.</p>
<div class="graph-block" 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" data-width="520" data-height="260" style="width:520px;height:260px;max-width:100%;display:block;margin:0 auto;"></div>
<p>The adventure park owner proposes to place the upper launch station at point $A(0, 75, 50)$, and the lower landing station at point $B(600, 25, 20)$. Points $C$ and $D$ are taken as the foot of perpendicular from points $B$ and $A$ onto the $x$-$y$ plane respectively.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-tjc-promo-q12-tjc-q12-item-block-3-item-0" data-list-item-path="h2-2025-tjc-promo-q12-tjc-q12-item-block-3-item-0"><div class="list-item-paragraph">Find a vector equation of line $AB$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-tjc-promo-q12-tjc-q12-item-block-3-item-1" data-list-item-path="h2-2025-tjc-promo-q12-tjc-q12-item-block-3-item-1"><div class="list-item-paragraph">Write down the coordinates of $C$ and $D$. Hence or otherwise, find the slope angle of descent, $\beta$, for the proposed zipline.<span class="inline-marks">[3]</span></div></li></ol>
<p>A viewing platform is to be constructed at a scenic point in the valley, located at point $P(105, 30, 30)$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="h2-2025-tjc-promo-q12-tjc-q12-item-block-5-item-0" data-list-item-path="h2-2025-tjc-promo-q12-tjc-q12-item-block-5-item-0"><div class="list-item-paragraph">Show that a vector equation of the plane $ABCD$ is $\mathbf{r}\cdot \begin{pmatrix} 1 \\ 12 \\ 0 \end{pmatrix}=900$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-tjc-promo-q12-tjc-q12-item-block-5-item-1" data-list-item-path="h2-2025-tjc-promo-q12-tjc-q12-item-block-5-item-1"><div class="list-item-paragraph">Find the exact coordinates of the foot of perpendicular from $P$ to the plane $ABCD$.<span class="inline-marks">[4]</span></div></li></ol>

An adventure park is designing a new zipline that will span between two hilltops. The diagram below shows a vertical cross-section of the zipline setup. Points are defined relative to a main control centre at $(0, 0, 0)$. The ground is horizontal and is modelled as the $x$-$y$ plane.

The adventure park owner proposes to place the upper launch station at point $A(0, 75, 50)$, and the lower landing station at point $B(600, 25, 20)$. Points $C$ and $D$ are taken as the foot of perpendicular from points $B$ and $A$ onto the $x$-$y$ plane respectively.

Write down the coordinates of $C$ and $D$. Hence or otherwise, find the slope angle of descent, $\beta$, for the proposed zipline.

A viewing platform is to be constructed at a scenic point in the valley, located at point $P(105, 30, 30)$.

Show that a vector equation of the plane $ABCD$ is $\mathbf{r}\cdot \begin{pmatrix} 1 \\ 12 \\ 0 \end{pmatrix}=900$.

Find the exact coordinates of the foot of perpendicular from $P$ to the plane $ABCD$.

Perpendicular Projection and Reflection in a Plane

Angle Between a Line and a Plane

Equations of Lines

Three-Dimensional Geometry

Equations of Planes

2025 TJC Promo Q12

Vectors

<p>With reference to the point $O$ as the origin and the $x$-$y$ plane as a horizontal plane, the pyramid $OPQRV$ has a parallelogram base $OPQR$ and height $OV$. The position vectors of the points $P$ and $R$ are $-3\mathbf{i} + 4\mathbf{j} + 3\mathbf{k}$ and $5\mathbf{i} - 2\mathbf{j} - \mathbf{k}$ respectively.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the coordinates of the point $S$ that lies on the line $PR$ such that the distance from $O$ to $S$ is a minimum.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the cartesian equations of the planes such that the perpendicular distance from each plane to the base $OPQR$ is $2\sqrt{86}$ units.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the acute angle between $OV$ and the vertical.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Given that&nbsp;$\overrightarrow{QV}$ is parallel to the vector $-5\mathbf{j} + 8\mathbf{k}$, find the position vector of the point $V$. Hence find the exact volume of the pyramid $OPQRV$.<span class="inline-marks">[5]</span></div></li></ol>
<p>$\text{[Volume of a pyramid} = \frac{1}{3} \times \text{base area} \times \text{height]}$</p>

With reference to the point $O$ as the origin and the $x$-$y$ plane as a horizontal plane, the pyramid $OPQRV$ has a parallelogram base $OPQR$ and height $OV$. The position vectors of the points $P$ and $R$ are $-3\mathbf{i} + 4\mathbf{j} + 3\mathbf{k}$ and $5\mathbf{i} - 2\mathbf{j} - \mathbf{k}$ respectively.

$\text{[Volume of a pyramid} = \frac{1}{3} \times \text{base area} \times \text{height]}$

Perpendicular Projection and Reflection in a Line

Distance from a Point to a Plane

Angles Between Two Lines

Intersection of Two Lines

Vectors in a Parallelogram

Areas of Triangles and Parallelograms Using Vectors

2025 VJC P2 Q3

<p>The position vectors of points $A$, $B$ and $C$, relative to the origin $O$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively, where</p>
<p style="text-align: center">\[\mathbf{a} = 3\mathbf{i} + 2\mathbf{j} - \mathbf{k},\quad \mathbf{b} = 4\mathbf{i} - 3\mathbf{j} + 2\mathbf{k},\quad \mathbf{c} = 3\mathbf{i} - \mathbf{j} - \mathbf{k}.\]</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Find $\mathbf{a} \times \mathbf{b}$ and deduce the exact area of triangle $OAB$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">Hence, find the volume of the tetrahedron $OABC$.</div><div class="list-item-paragraph">[The volume of a tetrahedron is $\frac{1}{3} \times \text{base area} \times \text{perpendicular height}$.]<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c"><div class="list-item-paragraph">Find the perpendicular distance from $B$ to line $OA$.<span class="inline-marks">[2]</span></div></li></ol>

The position vectors of points $A$, $B$ and $C$, relative to the origin $O$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively, where

\[\mathbf{a} = 3\mathbf{i} + 2\mathbf{j} - \mathbf{k},\quad \mathbf{b} = 4\mathbf{i} - 3\mathbf{j} + 2\mathbf{k},\quad \mathbf{c} = 3\mathbf{i} - \mathbf{j} - \mathbf{k}.\]

Vector (Cross) Product

Distance from a Point to a Line

2024 NYJC Promo Q5

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1775962953671-l5kobtsrm5n.jpg?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzU5NjI5NTM2NzEtbDVrb2J0c3JtNW4uanBnIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwMTY5NCwiZXhwIjoxNzkzNTAxOTk0fQ.RJIYFPqHh5RarT14vp3hEI0SZJ3HmhHcucoZUFHURkw" style="max-width: min(270px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>In the diagram, $O$ is the centre of the rectangular base $ABCD$ of a right pyramid with vertex $V$. Perpendicular unit vectors $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ are parallel to $AB$, $BC$ and $OV$ respectively. The length of $AB$, $BC$ and $OV$ are $12\text{ cm}$, $6\text{ cm}$ and $6\text{ cm}$ respectively. A line $l$ has Cartesian equation $\frac{-x-4}{10} = y+2 = \frac{t-z}{-2}.$</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find a vector equation of $AV$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">If the line $l$ intersects $AV$ at $M$, find the position vector of $M$ and the value of $t$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the acute angle between $AV$ and the line $l$. Hence find the perpendicular distance from $A$ to the line $l$.</span></div></li></ol>

In the diagram, $O$ is the centre of the rectangular base $ABCD$ of a right pyramid with vertex $V$. Perpendicular unit vectors $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ are parallel to $AB$, $BC$ and $OV$ respectively. The length of $AB$, $BC$ and $OV$ are $12\text{ cm}$, $6\text{ cm}$ and $6\text{ cm}$ respectively. A line $l$ has Cartesian equation $\frac{-x-4}{10} = y+2 = \frac{t-z}{-2}.$

If the line $l$ intersects $AV$ at $M$, find the position vector of $M$ and the value of $t$.

Find the acute angle between $AV$ and the line $l$. Hence find the perpendicular distance from $A$ to the line $l$.

2006 HCI P1 Q14

<p>To construct a special tentage, $3$ steel poles $OR$, $AQ$ and $BP$ are secured to the horizontal ground such that each of the poles are perpendicular to the ground. The point $O$ on the ground is taken as the origin and the unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OB$ and $OR$ respectively, with units in metres. It is given that&nbsp;$OR=12$, $AQ=9$, $BP=6$, $OA=20$ and $OB=15$ (see diagram).</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1771903221872-mq6u65oef5.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzE5MDMyMjE4NzItbXE2dTY1b2VmNS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTAxNjk0LCJleHAiOjE3OTM1MDE5OTR9.arPuD_yz1pVhQpK_LdkhRp4Pxz_EkeFrdpHwHFHh6kE" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find $\vec{PR}$ and $\vec{PQ}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph"><p>The canvas roof $PQR$, which is of negligible thickness, is part of the plane $p$. Using your answers in part (i), show that the equation of $p$ is given by ${\mathbf{r} \cdot \begin{pmatrix} 3 \\ 8 \\ 20 \end{pmatrix} = 240.}$</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Find the acute angle between the canvas roof and the plane $OAB$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">A cable is to be installed close to the tentage. This cable is part of the line $L$ with vector equation 	$\mathbf{r}=\left( \begin{matrix}
   \begin{matrix}
   -2  \\
   8  \\
   12  \\
\end{matrix}  \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
   \begin{matrix}
   20  \\
   13  \\
   -4  \\
\end{matrix}  \\
\end{matrix} \right)$, where&nbsp;$\lambda \in \mathbb{R}$. Show that this cable is not parallel to the edge $PR$ of the roof and does not meet 	the edge $PR$. <span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-v" data-list-item-path="item-v"><div class="list-item-paragraph">A lamp is hung vertically downwards from the canvas roof using a cable such that it is at the point $M$ with coordinates 	$\left( \frac{20}{3},5,8 \right)$. To secure the lamp, a second cable is connected from the lamp to the point $N$ on the roof. It is assumed 	that the cables are laid in straight lines and have negligible thickness. For the shortest length to be used for the second 	cable, show that&nbsp;$\overrightarrow{ON}=\left( \begin{matrix}
   \begin{matrix}
   s  \\
   \frac{2525}{473}  \\
   t  \\
\end{matrix}  \\
\end{matrix} \right)$, where&nbsp;$s$ and $t$ are constants to be found. <br><span class="inline-marks">[3]</span></div></li></ol>

To construct a special tentage, $3$ steel poles $OR$, $AQ$ and $BP$ are secured to the horizontal ground such that each of the poles are perpendicular to the ground. The point $O$ on the ground is taken as the origin and the unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OB$ and $OR$ respectively, with units in metres. It is given that $OR=12$, $AQ=9$, $BP=6$, $OA=20$ and $OB=15$ (see diagram).

Find the acute angle between the canvas roof and the plane $OAB$.

Vector Subtraction

Angle Between Two Planes

Relationships Between Two Lines

2024 MI PRU1 P1 Q9

<div class="graph-block" 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" data-width="215" data-height="136" data-proportional style="width:215px;height:136px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:215px;--graph-aspect:215 / 136;"></div>
<p>The diagram shows a cuboid with rectangular base $OABC$ and top $EFGH$, where&nbsp;$OA=4$ units, $OC=3$ units and $OE=2$ units. The point $O$ is taken as the origin and unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$, are taken along $OA$, $OC$ and $OE$ respectively.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find the cartesian equation of the plane $p$ which contains the points $A$, $C$ and $E$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find the acute angle between $p$ and the base $OABC$.<span class="inline-marks">[2]</span></div></li></ol>
<p>The line $l$, passing through $O$, is perpendicular to $p$ and intersects the plane containing $B$, $C$, $G$ and $H$ at the point $T$.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Find the position vector of the point $T$ and deduce the perpendicular distance from $T$ to $p$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">A point $Q$ lies on the line passing through $C$ and $T$ such that its distance from $p$ is twice that of the distance from $T$ to $p$. Find the possible position vectors of the point $Q$.<span class="inline-marks">[3]</span></div></li></ol>

The diagram shows a cuboid with rectangular base $OABC$ and top $EFGH$, where $OA=4$ units, $OC=3$ units and $OE=2$ units. The point $O$ is taken as the origin and unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$, are taken along $OA$, $OC$ and $OE$ respectively.

Find the cartesian equation of the plane $p$ which contains the points $A$, $C$ and $E$.

Find the acute angle between $p$ and the base $OABC$.

The line $l$, passing through $O$, is perpendicular to $p$ and intersects the plane containing $B$, $C$, $G$ and $H$ at the point $T$.

Find the position vector of the point $T$ and deduce the perpendicular distance from $T$ to $p$.

A point $Q$ lies on the line passing through $C$ and $T$ such that its distance from $p$ is twice that of the distance from $T$ to $p$. Find the possible position vectors of the point $Q$.

Intersection of a Line and a Plane

2013 DHS P1 Q12

<p>A designer wants to construct an inclined roof that is attached to a vertical rectangular wall, positioned above a horizontal flat ground, as shown in the diagram below. Assume that the roof and wall are of negligible thickness.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1768719798428-vepnc083dwo.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg3MTk3OTg0MjgtdmVwbmMwODNkd28ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwMTY5NCwiZXhwIjoxNzkzNTAxOTk0fQ.yEVUAKEOffgg9HAY8zGUgp-vrIPYh7d5WSV5T0k2TWc" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Points $(x, y, z)$ are defined relative to a corner of the wall at $O\,\mathrm (0, 0, 0)$, where units are metres. Unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are in the directions shown in the diagram. The other corners of the wall are $A$, $B$ and $C$, such that&nbsp;$AB$ is $5$ m and $A$ and $B$ are $3$ m vertically above $O$ and $C$ respectively. The roof is attached to the wall along $AB$ and is modelled by a plane parallel to $4\mathbf{i} + 5\mathbf{j} + \mathbf{k}$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that the roof is defined by the plane $x - 4z = -12$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the obtuse angle between the wall and the roof, correct to the nearest degree.<span class="inline-marks">[3]</span></div></li></ol>
<p>To support the roof, a wooden strut, of negligible thickness, is constructed. It has one end at point $P$ with coordinates $(4, 3, 0)$ on the horizontal ground and the other end at the roof, such that the length of the wooden strut is a minimum.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the exact length of the wooden strut.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">A ray of light is emitted in the direction $7\mathbf{i} - 4\mathbf{j} + 4\mathbf{k}$ from a light source at $C$. Show that this light ray meets the wooden strut. Hence, find the coordinates of the point where the light ray meets the wooden strut.<span class="inline-marks">[4]</span></div></li></ol>

A designer wants to construct an inclined roof that is attached to a vertical rectangular wall, positioned above a horizontal flat ground, as shown in the diagram below. Assume that the roof and wall are of negligible thickness.

Points $(x, y, z)$ are defined relative to a corner of the wall at $O\,\mathrm (0, 0, 0)$, where units are metres. Unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are in the directions shown in the diagram. The other corners of the wall are $A$, $B$ and $C$, such that $AB$ is $5$ m and $A$ and $B$ are $3$ m vertically above $O$ and $C$ respectively. The roof is attached to the wall along $AB$ and is modelled by a plane parallel to $4\mathbf{i} + 5\mathbf{j} + \mathbf{k}$.

Show that the roof is defined by the plane $x - 4z = -12$.

Find the obtuse angle between the wall and the roof, correct to the nearest degree.

To support the roof, a wooden strut, of negligible thickness, is constructed. It has one end at point $P$ with coordinates $(4, 3, 0)$ on the horizontal ground and the other end at the roof, such that the length of the wooden strut is a minimum.

Find the exact length of the wooden strut.

A ray of light is emitted in the direction $7\mathbf{i} - 4\mathbf{j} + 4\mathbf{k}$ from a light source at $C$. Show that this light ray meets the wooden strut. Hence, find the coordinates of the point where the light ray meets the wooden strut.

2025 JPJC P1 Q12

<p>The diagram shows a pyramid with horizontal rectangular base $ABCD$, where&nbsp;$AB=8$ cm, $BC=6$ cm. The vertex $V$ of the pyramid is vertically above the centre of the base and the length of $VA$ is $9$ cm. The points $E$, $F$ are the mid-points of $VA$, $VD$ respectively. The point $N$ is on $VB$ such that&nbsp;$VN:NB=2:1$.<br>Giving each answer correct to $3$ significant, figures, find</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--1-2" data-layout="1-2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li data-list-item-id="item-1780162733442-bmaxc36h6" data-list-item-path="item-1780162733442-bmaxc36h6"><div class="list-item-paragraph">the angle between plane $EFCB$ and the base</div></li><li data-list-item-id="item-1780162734386-ld2pqmao0" data-list-item-path="item-1780162734386-ld2pqmao0"><div class="list-item-paragraph">the angle which $CN$ makes with the base.</div></li></ol></div><div class="question-column" style="flex: 2; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="174" data-height="162" data-proportional style="width:174px;height:162px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:174px;--graph-aspect:174 / 162;"></div></div></div></div>

The diagram shows a pyramid with horizontal rectangular base $ABCD$, where $AB=8$ cm, $BC=6$ cm. The vertex $V$ of the pyramid is vertically above the centre of the base and the length of $VA$ is $9$ cm. The points $E$, $F$ are the mid-points of $VA$, $VD$ respectively. The point $N$ is on $VB$ such that $VN:NB=2:1$.
Giving each answer correct to $3$ significant, figures, find

Ratio Theorem and Section Formula

<p>A pyramid with a square base $ABCD$ is as shown in the diagram below. The dimensions are $AB=BC=CD=DA=4$ cm and $VA=VB=VC=VD=7$ cm. $N$ is a point at the foot of the perpendicular from $A$ to $VB$.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li data-list-item-id="item-1780162614868-yu01ylsps" data-list-item-path="item-1780162614868-yu01ylsps"><div class="list-item-paragraph">Show that the length of $VN$ is $5.86$ cm.</div></li><li data-list-item-id="item-1780162615818-isktwdaxd" data-list-item-path="item-1780162615818-isktwdaxd"><div class="list-item-paragraph">Find the angle between the plane $ANC$ and the horizontal.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="150" data-height="154" data-proportional style="width:150px;height:154px;max-width:100%;display:flow-root;margin:0 0 0 auto;--graph-width:150px;--graph-aspect:150 / 154;"></div></div></div></div>

A pyramid with a square base $ABCD$ is as shown in the diagram below. The dimensions are $AB=BC=CD=DA=4$ cm and $VA=VB=VC=VD=7$ cm. $N$ is a point at the foot of the perpendicular from $A$ to $VB$.

<p>The diagram below shows a triangular base pyramid with vertices, $A$, $B$, $C$ and $D$. With reference to the origin $O$, the points $A$, $B$, $C$ and $D$ are $\mathbf{i}+2\mathbf{j}+3\mathbf{k}$; $2\mathbf{i}-\mathbf{j}+4\mathbf{k}$; $4\mathbf{i}+5\mathbf{j}-\mathbf{k}$ and $-2\mathbf{i}-5\mathbf{k}$ respectively.</p>
<p>Let the plane containing points $A$, $B$ and $C$ be represented by $\Pi $.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761383152386-hccglh63i7j.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzODMxNTIzODYtaGNjZ2xoNjNpN2oucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwMTY5NCwiZXhwIjoxNzkzNTAxOTk0fQ.pr7Pp4EJVTkuJopUXouYJyNzK3afSOCEtziYYe9LORw" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-1761383164596-7z0soc7u7" data-list-item-path="item-1761383164596-7z0soc7u7"><div class="list-item-paragraph">Show that the equation of $\Pi $ can be expressed as&nbsp;$\mathbf{r}\cdot \left( \begin{matrix}
   \begin{matrix}
   9  \\
   7  \\
   12  \\
\end{matrix}  \\
\end{matrix} \right)=\alpha $, where&nbsp;$\alpha $ is a constant to be determined.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1761383165586-ami8jwbi1" data-list-item-path="item-1761383165586-ami8jwbi1"><div class="list-item-paragraph">Find the acute angle between line $BD$ and $\Pi $.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761383166209-tk9psry0s" data-list-item-path="item-1761383166209-tk9psry0s"><div class="list-item-paragraph">Find the position vector of the foot of perpendicular from the point $D$ to $\Pi $.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1761383166951-pqgx30t8k" data-list-item-path="item-1761383166951-pqgx30t8k"><div class="list-item-paragraph">Find the area of triangle $ABC$. Hence, find the volume of the pyramid.<span class="inline-marks">[4]</span></div></li></ol>
<p>$\left[ \mathrm{Volume of pyramid}=\frac{1}{3}\times \mathrm{base area}\times \mathrm{height} \right]$</p>

The diagram below shows a triangular base pyramid with vertices, $A$, $B$, $C$ and $D$. With reference to the origin $O$, the points $A$, $B$, $C$ and $D$ are $\mathbf{i}+2\mathbf{j}+3\mathbf{k}$; $2\mathbf{i}-\mathbf{j}+4\mathbf{k}$; $4\mathbf{i}+5\mathbf{j}-\mathbf{k}$ and $-2\mathbf{i}-5\mathbf{k}$ respectively.

Let the plane containing points $A$, $B$ and $C$ be represented by $\Pi $.

$\left[ \mathrm{Volume of pyramid}=\frac{1}{3}\times \mathrm{base area}\times \mathrm{height} \right]$

2024 CJC Promo Q12

<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761382105821-2y2g8mc3ico.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzODIxMDU4MjEtMnkyZzhtYzNpY28ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwMTY5NCwiZXhwIjoxNzkzNTAxOTk0fQ.VLKjuntBctJvM3IzntEyR9v3t9gIuaokTL0c0hNvW9s" width="400" class="aligncenter" /></figure>
<p>Methane $\left( \mathrm{C}{{\mathrm{H}}_{4}} \right)$ is a chemical compound with a tetrahedral structure. The $4$ hydrogen $\left( \mathrm{H} \right)$ atoms form a $4$ regular tetrahedron, and the carbon $\left( \mathrm{C} \right)$ atom is in the centre.</p>
<p>Let the centre of the $\mathrm{C}$-atom be the point $P$, and the centres of the $4$ $\mathrm{H}$-atoms be the points $Q$, $R$, $S$ and $T$. The coordinates of $P$, $Q$, $R$ and $S$ are $\left( -1,\,\,0,\,\,2 \right)$, $\left( -2,\,\,-1,\,\,1 \right)$, $\left( -2,\,\,1,\,\,3 \right)$ and $\left( 0,\,\,-1,\,\,\alpha \right)$ respectively.</p>
<p>The angle $\theta $ subtended by any two $\mathrm{C-H}$bonds at the $\mathrm{C}$-atom, such as angle $QPR$, is known as the $\mathrm{H-C-H}$ bond angle (see diagram above).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the bond angle, correct to $2$ decimal places.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">By using the fact that $QS=RS$, show that $a=3$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find a cartesian equation for plane $\pi $, which contains the points $P$, $Q$ and $R$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">$F$ is the point on $\pi $ that is closest to the point $S$.</div><ol class="i-list"><li><div class="list-item-paragraph">State a vector equation for the line $SF$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Hence, show that the coordinates of $F$ are $\left( 0,\,\,0,\,\,2 \right)$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Given that the point $T$ is the mirror image of the point $S$ in $\pi $, find the position vector of $T$.</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

Methane $\left( \mathrm{C}{{\mathrm{H}}_{4}} \right)$ is a chemical compound with a tetrahedral structure. The $4$ hydrogen $\left( \mathrm{H} \right)$ atoms form a $4$ regular tetrahedron, and the carbon $\left( \mathrm{C} \right)$ atom is in the centre.

Let the centre of the $\mathrm{C}$-atom be the point $P$, and the centres of the $4$ $\mathrm{H}$-atoms be the points $Q$, $R$, $S$ and $T$. The coordinates of $P$, $Q$, $R$ and $S$ are $\left( -1,\,\,0,\,\,2 \right)$, $\left( -2,\,\,-1,\,\,1 \right)$, $\left( -2,\,\,1,\,\,3 \right)$ and $\left( 0,\,\,-1,\,\,\alpha \right)$ respectively.

The angle $\theta $ subtended by any two $\mathrm{C-H}$bonds at the $\mathrm{C}$-atom, such as angle $QPR$, is known as the $\mathrm{H-C-H}$ bond angle (see diagram above).

Angles Between Vectors

Magnitude of a Vector

2023 EJC Promo Q12

<p>The diagram below shows a ${3}$-dimensional structure in which a pentahedron $DEFGMN$ lies on top of a trapezoidal prism $OABCDEFG$. Taking $O$ as the origin, perpendicular vectors $\mathbf{i}$ and $\mathbf{j}$ are parallel to $OA$ and $OC$ respectively. The base of the structure sits on the horizontal $x\mathrm{-}y$ plane.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761381586872-2zzhltlmfxc.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzODE1ODY4NzItMnp6aGx0bG1meGMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwMTY5NCwiZXhwIjoxNzkzNTAxOTk0fQ.0Vm84Q6J-LbYqZl-wSd_f4lA9Fr0rycGteyTHkJZblY" style="max-width: min(200px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Planes $OABC$ and $DEFG$ are parallel to each other. It is given that&nbsp;$OC$, $AB$, $DG$ and $EF$ are parallel to one another where&nbsp;$OC=AB=DG=EF=3$ units. It is also given that&nbsp;$OA$, $CB$, $DE$, $GF$ and $MN$ are parallel to one another where&nbsp;$OA=CB=MN=6$ units and $DE=GF=10$ units. The pentahedron $DEFGMN$ and the trapezoidal prism $OABCDEFG$ each has a height of $12$ units.</p>
<ol class="i-list"><li><div class="list-item-paragraph">The point $D$ has coordinates $\left( -2,0,s \right)$. State the value of $s$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">The line $DM$ is parallel to the vector $\left( \begin{matrix}
   1  \\
   1  \\
   t  \\
\end{matrix} \right)$ and the plane $ABFE$ has equation$6x=36+z$. It is given that 	the line $DM$ does not intersect with the plane $ABFE$. Find the value of $t$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Show that the equation of the plane $DGM$ is given by $\mathbf{r}\cdot \left( \begin{matrix}
   \begin{matrix}
   6  \\
   0  \\
   -1  \\
\end{matrix}  \\
\end{matrix} \right)=k$, where&nbsp;$k$ is a constant to be 	determined.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the acute angle between the planes $DGM$ and $DEFG$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the coordinates of $M$ and the exact shortest distance from $M$ to the plane $ABFE$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Another plane $\Pi $ has cartesian equation $x=c$, where&nbsp;$c$ is a constant. If the three planes $OAED$, $EFN$ 	and $\Pi $ all intersect at the point $E$, find the value of $c$, showing your working clearly.<span class="inline-marks">[2]</span></div></li></ol>

The diagram below shows a ${3}$-dimensional structure in which a pentahedron $DEFGMN$ lies on top of a trapezoidal prism $OABCDEFG$. Taking $O$ as the origin, perpendicular vectors $\mathbf{i}$ and $\mathbf{j}$ are parallel to $OA$ and $OC$ respectively. The base of the structure sits on the horizontal $x\mathrm{-}y$ plane.

Planes $OABC$ and $DEFG$ are parallel to each other. It is given that $OC$, $AB$, $DG$ and $EF$ are parallel to one another where $OC=AB=DG=EF=3$ units. It is also given that $OA$, $CB$, $DE$, $GF$ and $MN$ are parallel to one another where $OA=CB=MN=6$ units and $DE=GF=10$ units. The pentahedron $DEFGMN$ and the trapezoidal prism $OABCDEFG$ each has a height of $12$ units.

Finding Points on a Plane

Relationships Between a Line and a Plane

2022 HCI P2 Q5

<p>The diagram below shows the structure of an indoor farm to be built. Edges $OA$, $DC$, $IH$ and $EF$ are vertical, and edges $OA$, $AB$, $BC$ and $CD$ are equal to $EF$, $FG$, $GH$ and $HI$ respectively. Diagram is not drawn to scale.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761381261981-4howg7ycz4v.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzODEyNjE5ODEtNGhvd2c3eWN6NHYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwMTY5NCwiZXhwIjoxNzkzNTAxOTk0fQ.iGAf3NRalJbsAyBROnfP-16ovzqTe9kJZTBI6elosRM" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Taking $O$ as the origin and the vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ as unit vectors along the $OD$, $OE$ and $OA$ respectively, the coordinates of $A$ and $C$ are $\left( 0,0,20 \right)$ and $\left( 50,0,25 \right)$ respectively. All lengths are measured in metres.</p>
<ol class="i-list"><li data-list-item-id="item-1761381243717-vahzc7j0h" data-list-item-path="item-1761381243717-vahzc7j0h"><div class="list-item-paragraph">Given that&nbsp;$-\mathbf{i}+4\mathbf{k}$ is a vector perpendicular to the flat roof $ABGF$, and $\mathbf{i}+2\mathbf{k}$ is perpendicular to the flat roof $BCHG$, find the obtuse angle between the two roofs.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761381244235-kr5h542vi" data-list-item-path="item-1761381244235-kr5h542vi"><div class="list-item-paragraph">A humidity sensor is to be installed along the line $BG$, $50$ metres away from $B$. Find the coordinates of the location of the sensor.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1761381246500-aiuwe6ozn" data-list-item-path="item-1761381246500-aiuwe6ozn"><div class="list-item-paragraph">A metal cable is anchored on the ground outside the farm in front of $OABCD$, at a point $25$ metres away from $O$ along $OD$ and $10$ metres in front of $OD$. The cable is attached to a point on $AB$ to secure the roof $ABGF$. Find the length of the shortest cable needed.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1761381247107-lc6y5g18z" data-list-item-path="item-1761381247107-lc6y5g18z"><div class="list-item-paragraph">State an assumption for the above calculations to be valid.<span class="inline-marks">[1]</span></div></li></ol>

The diagram below shows the structure of an indoor farm to be built. Edges $OA$, $DC$, $IH$ and $EF$ are vertical, and edges $OA$, $AB$, $BC$ and $CD$ are equal to $EF$, $FG$, $GH$ and $HI$ respectively. Diagram is not drawn to scale.

Taking $O$ as the origin and the vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ as unit vectors along the $OD$, $OE$ and $OA$ respectively, the coordinates of $A$ and $C$ are $\left( 0,0,20 \right)$ and $\left( 50,0,25 \right)$ respectively. All lengths are measured in metres.

Line of Intersection of Two Planes

2018 RVHS P2 Q3

<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761377601079-ji5nzuh313l.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzNzc2MDEwNzktamk1bnp1aDMxM2wucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwMTY5NCwiZXhwIjoxNzkzNTAxOTk0fQ.M88p6lW5KybFneaqd6jDg9LAbkJUW5ced3fuKFUUCOs" width="400" class="aligncenter" /></figure>
<p>The diagram shows a prism with triangular ends $ABF$ and $DCE$. $AD$, $BC$ and $FE$ are parallel edges of the prism. The rectangular plane $ADEF$ has equation $x-2y=0$ and the rectangular base $ABCD$ has equation $3x-y+z=4$. The coordinates of $C$ and $F$ are $\left( 1,0,1 \right)$ and $\left( 4,2,0 \right)$ respectively.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the acute angle between the plane $ADEF$ and the line $CF$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the length of projection of vector $\overrightarrow{CF}$ on the base $ABCD$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the position vector of the point $P$ on the line passing through $A$ and $D$, which is closest to the mid-point of line segment $CF$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Find the Cartesian equation of the third rectangular plane $BCEF$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the distance between the two triangular ends of the prism.</div><p style="text-align: right;">[2]</p></li></ol>

The diagram shows a prism with triangular ends $ABF$ and $DCE$. $AD$, $BC$ and $FE$ are parallel edges of the prism. The rectangular plane $ADEF$ has equation $x-2y=0$ and the rectangular base $ABCD$ has equation $3x-y+z=4$. The coordinates of $C$ and $F$ are $\left( 1,0,1 \right)$ and $\left( 4,2,0 \right)$ respectively.

Projection Lengths

2022 ACJC J2 MYE Q5

<p>The diagram (not drawn to scale) shows the structure of a partially constructed building that is built on a horizontal ground. The building has a square base foundation of $7$ m in length. Points $O$, $A$, $B$ and $C$ are the corners of the foundation of the building. The building currently consists of three vertical pillars $OD$, $AE$ and $CF$ of heights $8$ m, $5$ m and $5$ m respectively. A canvas is currently attached at $D$, $E$ and $F$, forming a temporary shelter for the building. $O$ is taken as the origin and vectors $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$, each of length $1$ m, are taken along $\overrightarrow{OA}$, $\overrightarrow{OC}$ and $\overrightarrow{OD}$ respectively.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761377134885-p2eiy7gjncc.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzNzcxMzQ4ODUtcDJlaXk3Z2puY2MucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwMTY5NCwiZXhwIjoxNzkzNTAxOTk0fQ.UQM9wu7LziflnA27El7pMyfyP4091H6alY1NYKfU-zI" style="max-width: min(500px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-paragraph">Find a Cartesian equation of the plane that represents the canvas $DEF$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the acute angle which the canvas $DEF$ makes with the horizontal ground.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given that the canvas is to be extended along the plane $DEF$ till it touches the horizontal ground, explain why point $B$ will lie beneath the canvas.<span class="inline-marks">[2]</span></div></li></ol>
<p>A cement roof is to be built to replace the extended canvas. A vertical partition wall is also to be built such that it is $d$ m away from and parallel to the plane $ODFC$, where&nbsp;$0 &lt; d &lt; 7$.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Find the exact vector equation of the line where the roof meets the partition wall. Show your working clearly, leaving your answer in terms of $d$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">A lighting point, $P$, is to be placed on the roof such that it is closest to $B$. Find the position vector of $P$.<span class="inline-marks">[3]</span></div></li></ol>

The diagram (not drawn to scale) shows the structure of a partially constructed building that is built on a horizontal ground. The building has a square base foundation of $7$ m in length. Points $O$, $A$, $B$ and $C$ are the corners of the foundation of the building. The building currently consists of three vertical pillars $OD$, $AE$ and $CF$ of heights $8$ m, $5$ m and $5$ m respectively. A canvas is currently attached at $D$, $E$ and $F$, forming a temporary shelter for the building. $O$ is taken as the origin and vectors $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$, each of length $1$ m, are taken along $\overrightarrow{OA}$, $\overrightarrow{OC}$ and $\overrightarrow{OD}$ respectively.

A cement roof is to be built to replace the extended canvas. A vertical partition wall is also to be built such that it is $d$ m away from and parallel to the plane $ODFC$, where $0 < d < 7$.

Relationships Between a Point and a Plane

2017 NJC P2 Q4

<p>In the diagram, $O$ is centre of the square base $ABCD$ of a right pyramid, vertex $V$. Perpendicular unit vectors $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ are parallel to $AB$, $AD$, $OV$ respectively. The length of $AB$ is 4 units and the length of $OV$ is $2h$ units. $P$, $Q$, $M$ and $N$ are mid-points of $AB$, $BC$, $CV$ and $VA$ respectively. The point $O$ is taken as the origin for position vectors.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761370065226-1nk23hwyjjt.jpg?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzNzAwNjUyMjYtMW5rMjNod3lqanQuanBnIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwMTY5NCwiZXhwIjoxNzkzNTAxOTk0fQ.WzIBVwdhj5zwCYY7Mw3No0CbEbXTECaWKeiV0d0OMnE" width="400" class="aligncenter" /></figure>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the equation of the line $PM$ may be expressed as $\mathbf{r}=\left( \begin{matrix} 0 \\ -2 \\ 0 \\
\end{matrix} \right)+t\left( \begin{matrix} 1 \\ 3 \\ h \\
\end{matrix} \right)$, where $t$ is a parameter.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the equation for the line $QN$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Show that the lines $PM$ and $QN$ intersect, and that the position vector $\overrightarrow{OX}$ of their point of intersection is $\left( \begin{matrix}
\frac{1}{2} \\
-\frac{1}{2} \\
\frac{1}{2}h \\
\end{matrix} \right)$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Given that $OX$ is perpendicular to $VB$, find the value of $h$ and calculate the acute angle between $PM$ and $QN$, giving your answer correct to the nearest $0.1{}^\circ $.</div><p style="text-align: right;">[4]</p></li></ol>

In the diagram, $O$ is centre of the square base $ABCD$ of a right pyramid, vertex $V$. Perpendicular unit vectors $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ are parallel to $AB$, $AD$, $OV$ respectively. The length of $AB$ is 4 units and the length of $OV$ is $2h$ units. $P$, $Q$, $M$ and $N$ are mid-points of $AB$, $BC$, $CV$ and $VA$ respectively. The point $O$ is taken as the origin for position vectors.

Perpendicular Vectors Using the Scalar Product

N1994 P1 Q14

<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761369656764-1mb6w67t847.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzNjk2NTY3NjQtMW1iNnc2N3Q4NDcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwMTY5NCwiZXhwIjoxNzkzNTAxOTk0fQ.0oKwv1yyuRl1GHwvLjCWtkOgssJU3EA0OzhjcwNNA30" width="400" class="aligncenter" /></figure>
<p>The diagram above shows a simple shelter designed by an engineer. It consists of three vertical columns, $OE$, $AF$ and $BG$ with their bases on the horizontal ground. The point $O$ is taken as the origin and the points $E$, $F$ and $G$ have position vectors given respectively by $\mathbf{e}=6\mathbf{k}$, $\mathbf{f}=8\mathbf{j}+10\mathbf{k}$, and $\mathbf{g}=4\mathbf{i}+2\mathbf{j}+4\mathbf{k}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Given that the plane $EFG$ forms the roof of the shelter, show that the cartesian equation of the plane $EFG$ is $3x-2y+4z=24$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the angle between the plane $EFG$ and the horizontal ground.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">The engineer has to drill a hole on the roof of the shelter in order to install a lamp. Given that the lamp is vertically above a point on the ground with position vector $2\mathbf{i}+3\mathbf{j}$, find the vertical distance from the drilled hole to the horizontal ground.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">$M$ is a point that lies on the line segment $EF$. The engineer decided to construct a vertical partition that intersects plane $EFG$ along the line segment $GM$.Given that the partition has a cartesian equation of $x+2y=8$, find the vector equation for line $GM$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the shortest distance from $A$ to the vertical wall partition.</div><p style="text-align: right;">[3]</p></li></ol>

The diagram above shows a simple shelter designed by an engineer. It consists of three vertical columns, $OE$, $AF$ and $BG$ with their bases on the horizontal ground. The point $O$ is taken as the origin and the points $E$, $F$ and $G$ have position vectors given respectively by $\mathbf{e}=6\mathbf{k}$, $\mathbf{f}=8\mathbf{j}+10\mathbf{k}$, and $\mathbf{g}=4\mathbf{i}+2\mathbf{j}+4\mathbf{k}$.

2008 ACJC P2 Q5

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761369383237-73ep5wlicp.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzNjkzODMyMzctNzNlcDV3bGljcC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTAxNjk0LCJleHAiOjE3OTM1MDE5OTR9.zzX6r_7yMz8E-bnuUcMKm8fAX9wmLhwCd8H_p4VDIgY" style="max-width: min(500px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The diagram shows a roof structure modelled by a prism* $ABCDEF$ with parallelogram sides. The triangular ends $ABF$ and $DCE$ are similar and parallel to the $xz$ plane. The origin $O$ is a point on the horizontal ground which is parallel to the $xy$ plane. The coordinates of point $B$ and $M$ are $\left( 2,-2,11 \right)$ and $\left( -1.75,-2,8 \right)$ respectively. It is given that plane $ABCD$ has equation $48x+5y-60z=-574$and plane $BCEF$ has equation $12x-y+12z=158$.</p>
<p style="font-size: 0.75rem"><p style="color: red;">*While the diagram models a roof structure using a prism, there appears to be a conceptual misunderstanding in the question. A prism, by definition, is a three-dimensional geometric shape with two parallel, congruent bases connected by rectangular or parallelogram faces, which implies a constant cross-sectional area along its length. However, in this case, the structure described seems to have varying cross-sectional areas, which does not conform to the standard geometric properties of a prism. Thus, this structure is **not a prism**.</p></p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the acute angle between plane $ABCD$ and plane $BCEF$, giving your answer correct to $2$ decimal places.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Without the use of the calculator, show that the equation of line $BC$ is $x=2$, $\frac{y+2}{12}=z-11$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">It is now given that the vertical height of point $C$ from the horizontal ground is $11.25$ metres. State the coordinates of point $C$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the acute angle between the line $BC$ and the horizontal ground, giving your answer correct to $2$ decimal places.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given that point $N$ lies on $BF$ such that&nbsp;$MN$ is parallel to the $x$-axis, find the length of $MN$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">To strengthen the roof structure and minimise material cost, a steel beam $MW$ is constructed such that point $W$ lies on plane $BCEF$ and is closest to point $M$. Find the length of the steel beam $MW$.<span class="inline-marks">[3]</span></div></li></ol>

The diagram shows a roof structure modelled by a prism* $ABCDEF$ with parallelogram sides. The triangular ends $ABF$ and $DCE$ are similar and parallel to the $xz$ plane. The origin $O$ is a point on the horizontal ground which is parallel to the $xy$ plane. The coordinates of point $B$ and $M$ are $\left( 2,-2,11 \right)$ and $\left( -1.75,-2,8 \right)$ respectively. It is given that plane $ABCD$ has equation $48x+5y-60z=-574$and plane $BCEF$ has equation $12x-y+12z=158$.

<p style="color: red;">*While the diagram models a roof structure using a prism, there appears to be a conceptual misunderstanding in the question. A prism, by definition, is a three-dimensional geometric shape with two parallel, congruent bases connected by rectangular or parallelogram faces, which implies a constant cross-sectional area along its length. However, in this case, the structure described seems to have varying cross-sectional areas, which does not conform to the standard geometric properties of a prism. Thus, this structure is **not a prism**.</p>

2023 ACJC Promo Q11

<p>At a theme park, the management decides to introduce a zipline adventure where a person can travel down from the top to the bottom of an inclined zipline on a pulley system, above a flat horizontal field.</p>
<p>Points are defined relative to an origin $\left( 0,\mathrm{ }0,\mathrm{ }0 \right)$ on the flat horizontal field, where units are metres. The $x$-, $y$- and $z$-axes are in the directions of east, north and vertically upwards respectively. The zipline is above a flat horizontal field modelled by the equation $z=0$. A take-off point is located at $\left( 50,\mathrm{ }50,\mathrm{ }100 \right)$ on a vertical tower and an anchor point lies on another vertical tower as shown in the diagram below. The zipline is assumed to be taut at all times and the width of the zipline can be neglected.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761367242442-gguxzknrhck.jpg?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzNjcyNDI0NDItZ2d1eHprbnJoY2suanBnIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwMTY5NCwiZXhwIjoxNzkzNTAxOTk0fQ.1-qFjzxJnsIuiWqne0t89Jc6P7sr6WIePUpiaaKBM7k" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-1761367276952-lfly688cg" data-list-item-path="item-1761367276952-lfly688cg"><div class="list-item-paragraph">The anchor point lies on a cable modelled by the equation $\mathbf{r}=\left( \begin{matrix}
25 \\
20 \\
20 \\
\end{matrix} \right)+\mu \left( \begin{matrix}
2 \\
1 \\
2 \\
\end{matrix} \right)$, $\mu \in \mathbb{R}$. Given that the zipline is $100$m long with one end attached to the take-off point, find the coordinates of the anchor point, giving your answer correct to $2$ decimal places.<span class="inline-marks">[4]</span></div></li></ol>
<p>Due to safety reasons, the theme park management decides to relocate the anchor point and the new zipline follows an equation $\mathbf{r}=\left( \begin{matrix}
50 \\
50 \\
100 \\
\end{matrix} \right)+\beta \left( \begin{matrix}
3 \\
0 \\
1 \\
\end{matrix} \right)$, $\beta \in \mathbb{R}$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1761367322596-00evvurny" data-list-item-path="item-1761367322596-00evvurny"><div class="list-item-paragraph">Find the length of projection of the new zipline onto the field, given that the new zipline is also $100$m long.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1761367323279-fqejn4qde" data-list-item-path="item-1761367323279-fqejn4qde"><div class="list-item-paragraph">A flat safety net is to be constructed below the new zipline such that it is parallel to both the new zipline and the vector $\left( \begin{matrix}
1 \\
3 \\
4 \\
\end{matrix} \right)$. At the same time, the flat safety net is to keep a distance of $5$m from the new zipline at all times. Find the cartesian equation modelled by the flat safety net.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1761367324396-ktk893mxr" data-list-item-path="item-1761367324396-ktk893mxr"><div class="list-item-paragraph">If the zipline can be modelled by the equation $\mathbf{r}=\left( \begin{matrix}
50 \\
50 \\
100 \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
a \\
0 \\
c \\
\end{matrix} \right)$, $\lambda \in \mathbb{R}$, what can be said about the values of $\lambda $, $a$ and $c$?<span class="inline-marks">[2]</span></div></li></ol>

At a theme park, the management decides to introduce a zipline adventure where a person can travel down from the top to the bottom of an inclined zipline on a pulley system, above a flat horizontal field.

Points are defined relative to an origin $\left( 0,\mathrm{ }0,\mathrm{ }0 \right)$ on the flat horizontal field, where units are metres. The $x$-, $y$- and $z$-axes are in the directions of east, north and vertically upwards respectively. The zipline is above a flat horizontal field modelled by the equation $z=0$. A take-off point is located at $\left( 50,\mathrm{ }50,\mathrm{ }100 \right)$ on a vertical tower and an anchor point lies on another vertical tower as shown in the diagram below. The zipline is assumed to be taut at all times and the width of the zipline can be neglected.

Due to safety reasons, the theme park management decides to relocate the anchor point and the new zipline follows an equation $\mathbf{r}=\left( \begin{matrix}
50 \\
50 \\
100 \\
\end{matrix} \right)+\beta \left( \begin{matrix}
3 \\
0 \\
1 \\
\end{matrix} \right)$, $\beta \in \mathbb{R}$.

2023 NYJC P1 Q10

<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761365257856-bwiph4eca4.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzNjUyNTc4NTYtYndpcGg0ZWNhNC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTAxNjk0LCJleHAiOjE3OTM1MDE5OTR9.F0gfaVZFc_1X6Odn6q7bTuruFU6PPeKchcRkA2pSS-o" style="max-width: 300px; height: auto; display: block;" class="aligncenter" /></figure>
<p>The diagram above shows part of the structure of a rock-climbing wall, with a horizontal base $OABC$, a vertical rectangular wall $OADE$ and an inclined rectangular wall $DEFG$. With respect to the origin $O$, the vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ each of length $1$ m, are parallel to $OA$, $OC$ and $OE$ respectively. It is given that $OE=10$m.</p>
<p>The plane $OADE$ has equation $\mathbf{r}\cdot \left( \begin{matrix}
0 \\
1 \\
0 \\
\end{matrix} \right)=p$, and the plane $DEFG$ has equation $\mathbf{r}=\left( \begin{matrix}
0 \\
0 \\
10 \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
3 \\
-2 \\
4 \\
\end{matrix} \right)+\mu \left( \begin{matrix}
0 \\
-1 \\
q \\
\end{matrix} \right)$, where $p$ and $q$ are constants, $\lambda $ and $\mu $ are parameters.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Give a reason why $p=0$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Write down a vector equation of the line passing through the points $D$ and $E$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Show that $q=2$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence, find the obtuse angle between the walls $OADE$ and $DEFG$.<span class="inline-marks">[3]</span></div></li></ol>
<p>A rock climber is at the point $H\left( 2,\,-1,\,12 \right)$.</p>
<ol class="abc-list" start="5"><li><div class="list-item-paragraph">Show that the climber is on the plane $DEFG$.<span class="inline-marks">[2]</span></div></li></ol>
<p>The line $l$ passing through the points $F$ and $G$ has equation $\mathbf{r}=\left( \begin{matrix}
3 \\
-2 \\
14 \\
\end{matrix} \right)+\gamma \left( \begin{matrix}
1 \\
0 \\
0 \\
\end{matrix} \right)$, where $\gamma $ is a parameter.</p>
<ol class="abc-list" start="6"><li><div class="list-item-paragraph">Find the exact shortest distance from the climber to $l$.<span class="inline-marks">[3]</span></div></li></ol>

The diagram above shows part of the structure of a rock-climbing wall, with a horizontal base $OABC$, a vertical rectangular wall $OADE$ and an inclined rectangular wall $DEFG$. With respect to the origin $O$, the vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ each of length $1$ m, are parallel to $OA$, $OC$ and $OE$ respectively. It is given that $OE=10$m.

The plane $OADE$ has equation $\mathbf{r}\cdot \left( \begin{matrix}
0 \\
1 \\
0 \\
\end{matrix} \right)=p$, and the plane $DEFG$ has equation $\mathbf{r}=\left( \begin{matrix}
0 \\
0 \\
10 \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
3 \\
-2 \\
4 \\
\end{matrix} \right)+\mu \left( \begin{matrix}
0 \\
-1 \\
q \\
\end{matrix} \right)$, where $p$ and $q$ are constants, $\lambda $ and $\mu $ are parameters.

A rock climber is at the point $H\left( 2,\,-1,\,12 \right)$.

The line $l$ passing through the points $F$ and $G$ has equation $\mathbf{r}=\left( \begin{matrix}
3 \\
-2 \\
14 \\
\end{matrix} \right)+\gamma \left( \begin{matrix}
1 \\
0 \\
0 \\
\end{matrix} \right)$, where $\gamma $ is a parameter.

Math Question Bank