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N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>With reference to the origin $O$, the coordinates of $A$, $B$ and $C$ are $(2,8,-a)$, $(1,-10,3)$ and $(5,6,a)$ respectively. The plane $\pi$ contains points $O$, $A$, $B$ and $C$.</p>
<ol class="abc-list"><li data-list-item-id="ri-q11-l2-item01" data-list-item-path="ri-q11-l2-item01"><div class="list-item-paragraph">Show that&nbsp;$a=1$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="ri-q11-l2-item02" data-list-item-path="ri-q11-l2-item02"><div class="list-item-paragraph">Find the acute angle between $\pi$ and the $yz$-plane.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="ri-q11-l2-item03" data-list-item-path="ri-q11-l2-item03"><div class="list-item-paragraph">$\pi$ and the $yz$-plane meet in the line $l$. By evaluating an appropriate cross product, find a vector equation for&nbsp;$l$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="ri-q11-l2-item04" data-list-item-path="ri-q11-l2-item04"><div class="list-item-paragraph">Write down the distance from $A$ to the $yz$-plane. Hence or otherwise, find the distance from $A$ to $l$.<span class="inline-marks">[3]</span></div></li></ol>

With reference to the origin $O$, the coordinates of $A$, $B$ and $C$ are $(2,8,-a)$, $(1,-10,3)$ and $(5,6,a)$ respectively. The plane $\pi$ contains points $O$, $A$, $B$ and $C$.

Find the acute angle between $\pi$ and the $yz$-plane.

$\pi$ and the $yz$-plane meet in the line $l$. By evaluating an appropriate cross product, find a vector equation for $l$.

Write down the distance from $A$ to the $yz$-plane. Hence or otherwise, find the distance from $A$ to $l$.

Relationships Between a Point and a Plane

Distance from a Point to a Plane

Equations of Planes

Angle Between Two Planes

Line of Intersection of Two Planes

Distance from a Point to a Line

2026 RI P1 Q11

Vectors

<p>Relative to the origin $O$, the position vectors of $A$, $B$ and $C$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively where&nbsp;$\mathbf{c}=3\mathbf{b}-\mathbf{a}$. Given that&nbsp;$\mathbf{b}$ is a unit vector, $|\mathbf{a}|=3$, $|\mathbf{b}-\mathbf{a}|=\sqrt7$ and the angle between $\mathbf{a}$ and $\mathbf{b}$ is $\frac\pi3$,</p>
<ol class="i-list"><li data-list-item-id="asrjc-q06-item01" data-list-item-path="asrjc-q06-item01"><div class="list-item-paragraph">find the exact shortest distance from $C$ to the line $AB$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="asrjc-q06-item02" data-list-item-path="asrjc-q06-item02"><div class="list-item-paragraph">Find the angle between $\mathbf{b}$ and $\overrightarrow{AC}$.<span class="inline-marks">[3]</span></div></li></ol>

Relative to the origin $O$, the position vectors of $A$, $B$ and $C$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively where $\mathbf{c}=3\mathbf{b}-\mathbf{a}$. Given that $\mathbf{b}$ is a unit vector, $|\mathbf{a}|=3$, $|\mathbf{b}-\mathbf{a}|=\sqrt7$ and the angle between $\mathbf{a}$ and $\mathbf{b}$ is $\frac\pi3$,

find the exact shortest distance from $C$ to the line $AB$.

Find the angle between $\mathbf{b}$ and $\overrightarrow{AC}$.

Angles Between Vectors

2026 ASRJC P1 Q6

<p>With reference to the origin $O$, the points $A$, $B$ and $X$ have position vectors $3\mathbf{i}+2\mathbf{j}+\mathbf{k}$, $\mathbf{i}+2\mathbf{j}+3\mathbf{k}$ and $11\mathbf{i}+10\mathbf{j}+9\mathbf{k}$ respectively. The point $C$ lies on $OA$ produced such that&nbsp;$OA:AC=1:\alpha$, where&nbsp;$\alpha$ is a constant. The point $D$ lies on $OB$ produced such that&nbsp;$OB:BD=1:\beta$, where&nbsp;$\beta$ is a constant.</p>
<p>It is given that&nbsp;$OCXD$ is a parallelogram.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-tjc-promo-q3-tjc-q3-item-block-2-item-0" data-list-item-path="h2-2025-tjc-promo-q3-tjc-q3-item-block-2-item-0"><div class="list-item-paragraph">Write down the position vectors of $C$ and $D$ in terms of $\alpha$ and $\beta$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-tjc-promo-q3-tjc-q3-item-block-2-item-1" data-list-item-path="h2-2025-tjc-promo-q3-tjc-q3-item-block-2-item-1"><div class="list-item-paragraph">Show that&nbsp;$\beta=1$ and find the value of $\alpha$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-tjc-promo-q3-tjc-q3-item-block-2-item-2" data-list-item-path="h2-2025-tjc-promo-q3-tjc-q3-item-block-2-item-2"><div class="list-item-paragraph">Find the exact shortest distance from $X$ to line $OD$.<span class="inline-marks">[3]</span></div></li></ol>

With reference to the origin $O$, the points $A$, $B$ and $X$ have position vectors $3\mathbf{i}+2\mathbf{j}+\mathbf{k}$, $\mathbf{i}+2\mathbf{j}+3\mathbf{k}$ and $11\mathbf{i}+10\mathbf{j}+9\mathbf{k}$ respectively. The point $C$ lies on $OA$ produced such that $OA:AC=1:\alpha$, where $\alpha$ is a constant. The point $D$ lies on $OB$ produced such that $OB:BD=1:\beta$, where $\beta$ is a constant.

It is given that $OCXD$ is a parallelogram.

Write down the position vectors of $C$ and $D$ in terms of $\alpha$ and $\beta$.

Show that $\beta=1$ and find the value of $\alpha$.

Find the exact shortest distance from $X$ to line $OD$.

Vectors in a Parallelogram

Ratio Theorem and Section Formula

2025 TJC Promo Q3

<p>The planes $\pi_1$ and $\pi_2$, which meet in the line $l_1$, have equations</p>
<p style="text-align: center">\[\pi_1 : \mathbf{r} \cdot \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix} = 25 \quad \text{and} \quad \pi_2 : x + ky - 2z = -15,\]</p>
<p>where&nbsp;$k$ is a constant.</p>
<p>Another line $l_2$ has equation $\mathbf{r} = \begin{pmatrix} -15 \\ 0 \\ 0 \end{pmatrix} + \beta \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix}, \beta \in \mathbb{R}$.</p>
<ol class="i-list"><li data-list-item-id="imported-item-block-4-item-0" data-list-item-path="imported-item-block-4-item-0"><div class="list-item-paragraph">Determine the position vector of the point on $l_1$ such that the coordinates of this point is independent of $k$. Hence find a vector equation of $l_1$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="imported-item-block-4-item-1" data-list-item-path="imported-item-block-4-item-1"><div class="list-item-paragraph">Determine the possible value(s) of $k$ such that&nbsp;$l_1$ and $l_2$ are skew.<span class="inline-marks">[3]</span></div></li></ol>
<p>Assume that&nbsp;$k = 4$ for the rest of this question.</p>
<ol class="i-list" start="3"><li data-list-item-id="imported-item-block-6-item-0" data-list-item-path="imported-item-block-6-item-0"><div class="list-item-paragraph">Points $A$ and $B$ are on $l_1$ and $l_2$ respectively such that&nbsp;$\overrightarrow{AB}$ is perpendicular to both lines. Show that&nbsp;$\left\vert{} \overrightarrow{AB} \right\vert{} = \sqrt{\frac{p}{2}}$, where&nbsp;$p$ is an integer to be determined.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-6-item-1" data-list-item-path="imported-item-block-6-item-1"><div class="list-item-paragraph">Find exactly the sine of the acute angle between $l_2$ and $\pi_1$.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is given further that&nbsp;$l_2$ lies on a third plane that is perpendicular to $l_1$, and $l_2$ intersects $\pi_1$ at point $P$.</p>
<ol class="i-list" start="5"><li data-list-item-id="imported-item-block-8-item-0" data-list-item-path="imported-item-block-8-item-0"><div class="list-item-paragraph">Deduce the shortest distance from $P$ to $l_1$.<span class="inline-marks">[2]</span></div></li></ol>

The planes $\pi_1$ and $\pi_2$, which meet in the line $l_1$, have equations

\[\pi_1 : \mathbf{r} \cdot \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix} = 25 \quad \text{and} \quad \pi_2 : x + ky - 2z = -15,\]

Another line $l_2$ has equation $\mathbf{r} = \begin{pmatrix} -15 \\ 0 \\ 0 \end{pmatrix} + \beta \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix}, \beta \in \mathbb{R}$.

Determine the position vector of the point on $l_1$ such that the coordinates of this point is independent of $k$. Hence find a vector equation of $l_1$.

Determine the possible value(s) of $k$ such that $l_1$ and $l_2$ are skew.

Assume that $k = 4$ for the rest of this question.

Points $A$ and $B$ are on $l_1$ and $l_2$ respectively such that $\overrightarrow{AB}$ is perpendicular to both lines. Show that $\left\vert{} \overrightarrow{AB} \right\vert{} = \sqrt{\frac{p}{2}}$, where $p$ is an integer to be determined.

Find exactly the sine of the acute angle between $l_2$ and $\pi_1$.

It is given further that $l_2$ lies on a third plane that is perpendicular to $l_1$, and $l_2$ intersects $\pi_1$ at point $P$.

Deduce the shortest distance from $P$ to $l_1$.

Finding Points on a Plane

Angle Between a Line and a Plane

Common Perpendicular and Distance Between Skew Lines

Relationships Between Two Lines

Perpendicular Vectors Using the Scalar Product

2024 NJC P1 Q12

<p>The planes $p$ and $q$ have equations</p>
<p style="text-align: center">\[\mathbf{r} = (2 + \lambda - 2\mu)\mathbf{i} + (-3\lambda + \mu)\mathbf{j} + (3 - \lambda + 2\mu)\mathbf{k} \quad \text{and} \quad \mathbf{r} \cdot \begin{pmatrix} a \\ -1 \\ b \end{pmatrix} = 1 \text{ respectively,}\]</p>
<p>where&nbsp;$a$ and $b$ are constants and $\lambda$ and $\mu$ are parameters.</p>
<p>The line $l$ passes through the point $(5, 4, 0)$ and is parallel to the vector $-2\mathbf{i} - \mathbf{j} + 2\mathbf{k}$. The planes $p$ and $q$ meet in the line $l$.</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Show that&nbsp;$a = 1$ and $b = \frac{1}{2}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">Find the exact acute angle between the planes $p$ and $q$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c"><div class="list-item-paragraph">Find the distance from the point $A(2, 0, 3)$ to the plane $q$. Hence deduce the shortest distance from $A$ to $l$.<span class="inline-marks">[4]</span></div></li></ol>
<p>The plane $q$ is reflected about the plane $p$ to obtain the plane $q'$.</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-d"><div class="list-item-paragraph">Find a cartesian equation of the plane $q'$.<span class="inline-marks">[3]</span></div></li></ol>

The planes $p$ and $q$ have equations

\[\mathbf{r} = (2 + \lambda - 2\mu)\mathbf{i} + (-3\lambda + \mu)\mathbf{j} + (3 - \lambda + 2\mu)\mathbf{k} \quad \text{and} \quad \mathbf{r} \cdot \begin{pmatrix} a \\ -1 \\ b \end{pmatrix} = 1 \text{ respectively,}\]

where $a$ and $b$ are constants and $\lambda$ and $\mu$ are parameters.

The line $l$ passes through the point $(5, 4, 0)$ and is parallel to the vector $-2\mathbf{i} - \mathbf{j} + 2\mathbf{k}$. The planes $p$ and $q$ meet in the line $l$.

The plane $q$ is reflected about the plane $p$ to obtain the plane $q'$.

Perpendicular Projection and Reflection in a Plane

Relationships Between a Line and a Plane

2024 RI Promo Q9

<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

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

Three-Dimensional Geometry

Vector (Cross) Product

Areas of Triangles and Parallelograms Using Vectors

2024 NYJC Promo Q5

<p>In triangle $PQR$, the vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ represent the vectors $\overrightarrow{PQ}$, $\overrightarrow{QR}$ and $\overrightarrow{PR}$ 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/pasted/1779426366202-yu4qd1rzjpn.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Nzk0MjYzNjYyMDIteXU0cWQxcnpqcG4ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTc3MywiZXhwIjoxNzkyNjk2MDczfQ.TPtgR5vOWYQq0MdSr9oeV2gma3eoxcmbYdUqGoeXNMA" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">State a geometrical interpretation of $|\mathbf{c} \times \mathbf{\hat{a}}|$ and hence show that the area of the triangle $PQR$ can be written as&nbsp;$\frac{1}{2}|\mathbf{c} \times \mathbf{{a}}|$ units$^2$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">With clear explanations, show that&nbsp;$|\mathbf{c} \times \mathbf{a}| = |\mathbf{b} \times \mathbf{a}|$ and deduce that&nbsp;$\frac{|\mathbf{c}|}{\sin \angle PQR} = \frac{|\mathbf{b}|}{\sin \angle QPR}$ (Sine Rule).<span class="inline-marks">[2]</span></div></li></ol>

In triangle $PQR$, the vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ represent the vectors $\overrightarrow{PQ}$, $\overrightarrow{QR}$ and $\overrightarrow{PR}$ respectively.

2024 ASRJC Promo Q5

<p>An airfield on a horizontal flat piece of ground must manage the take-off flight paths of aircraft from its runways. These flight paths may be assumed to be straight lines, and the aircraft may be assumed to be point objects.</p>
<p>In the diagram below, the labelled points are defined relative to an origin $O$ with coordinates $(0,0,0)$ on the airfield. The $x$-, $y$- and $z$-axes are in the directions east, north and vertically upwards respectively, with units in metres. The direction vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ correspond to the east, north and vertically upwards directions respectively. The point $A$ has coordinates $(2000,500,0)$ and the point $B$ has coordinates $(1400,2900,0)$.</p>
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<p>Aircraft Alpha takes off from Point $A$ in the direction $4\mathbf{i} + 3\mathbf{j} + \mathbf{k}$ as shown.</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 initial take-off angle $\theta$, the angle that Aircraft Alpha's flight path makes with the horizontal ground.<span class="inline-marks">[2]</span></div></li></ol>
<p>The peak of a majestic mountain has coordinates $(18000,4500,2000)$.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find the coordinates of the closest point on the flight path of Aircraft Alpha to the peak. Hence find the closest distance between the flight path and the peak, giving your answer to the nearest metre.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Aircraft Bravo takes off from Point $B$. For aviation safety reasons, the flight paths of Aircraft Alpha and Aircraft Bravo should never cross each other. Given that Aircraft Bravo will take off in the direction $4\mathbf{i} + 2\mathbf{j} + h\mathbf{k}$, find the value of $h$ which will result in the flight paths crossing each other, giving your answer to $3$ decimal places.<span class="inline-marks">[2]</span></div></li></ol>
<p>The value of $h$ is now given to be $1$.</p>
<ol class="i-list" start="4"><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">Find the cartesian equation of the plane containing point $B$ and the flight path of Aircraft Alpha. Using this answer, provide another mathematical explanation as to why the flight path of Aircraft Bravo will not cross the flight path of Aircraft Alpha.<span class="inline-marks">[4]</span></div></li></ol>

An airfield on a horizontal flat piece of ground must manage the take-off flight paths of aircraft from its runways. These flight paths may be assumed to be straight lines, and the aircraft may be assumed to be point objects.

In the diagram below, the labelled points are defined relative to an origin $O$ with coordinates $(0,0,0)$ on the airfield. The $x$-, $y$- and $z$-axes are in the directions east, north and vertically upwards respectively, with units in metres. The direction vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ correspond to the east, north and vertically upwards directions respectively. The point $A$ has coordinates $(2000,500,0)$ and the point $B$ has coordinates $(1400,2900,0)$.

Aircraft Alpha takes off from Point $A$ in the direction $4\mathbf{i} + 3\mathbf{j} + \mathbf{k}$ as shown.

The peak of a majestic mountain has coordinates $(18000,4500,2000)$.

The value of $h$ is now given to be $1$.

Perpendicular Projection and Reflection in a Line

Intersection of Two Lines

2023 ACJC P1 Q12

<p>If&nbsp;$|\mathbf{a} \times \mathbf{b}| = 16$ and $|\mathbf{b} - \mathbf{a}| = 10$, find the shortest distance from $O$ to $AB$.<span class="inline-marks">[2]</span></p>

If $|\mathbf{a} \times \mathbf{b}| = 16$ and $|\mathbf{b} - \mathbf{a}| = 10$, find the shortest distance from $O$ to $AB$.

<p>The line $l_1$ has equation $x - 1 = 2 - y = -z$.</p>
<p>The line $l_2$ passes through the point $A(-2, 1, 7)$ and is parallel to the vector $-3\mathbf{i} + \mathbf{j} + 5\mathbf{k}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down vector equations for&nbsp;$l_1$ and $l_2$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that the lines $l_1$ and $l_2$ intersect and find the coordinates of $E$, the point of intersection of $l_1$ and $l_2$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">The acute angle between the lines $l_1$ and $l_2$ is denoted by $\theta$. By finding $\cos \theta$, show that \[\sin \theta = \frac{2\sqrt{2}}{\sqrt{35}}.\] Hence find the shortest distance from $A$ to the line $l_1$, leaving your answer in exact form.</span><span class="inline-marks">[5]</span></div></li></ol>

The line $l_1$ has equation $x - 1 = 2 - y = -z$.

The line $l_2$ passes through the point $A(-2, 1, 7)$ and is parallel to the vector $-3\mathbf{i} + \mathbf{j} + 5\mathbf{k}$.

Write down vector equations for $l_1$ and $l_2$.

Show that the lines $l_1$ and $l_2$ intersect and find the coordinates of $E$, the point of intersection of $l_1$ and $l_2$.

The acute angle between the lines $l_1$ and $l_2$ is denoted by $\theta$. By finding $\cos \theta$, show that \[\sin \theta = \frac{2\sqrt{2}}{\sqrt{35}}.\] Hence find the shortest distance from $A$ to the line $l_1$, leaving your answer in exact form.

Angles Between Two Lines

Equations of Lines

<p>The planes $p_1$, $p_2$ and $p_3$ have equations \[p_1 : -x + 2y + 4z = 5,\]\[p_2 : 3x + ay - 6z = 6,\]\[p_3 : 2x - 4y - 8z = b,\]where&nbsp;$a$ and $b$ are real constants, $b &gt; 0$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">The line $l$ has equation $\frac{x - 5}{8} = \frac{3 - y}{6} = \frac{z - 1}{5}$. Given that the planes $p_1$ and $p_2$ intersect at $l$, show that&nbsp;$a = -1$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the acute angle between $p_1$ and $p_2$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">The plane $p_4$ has equation $\mathbf{r} \cdot \begin{pmatrix} 1 \\ -2 \\ -4 \end{pmatrix} = 0$. Given that&nbsp;$p_1$ and $p_3$ are equidistant from $p_4$, find the value of $b$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">The point $A$ with coordinates $(-1, 0, 1)$ lies on $p_1$, and $F$ is the foot of the perpendicular from $A$ to $l$. The point $B$ is the reflection of $F$ in the plane $p_3$. Find the length of $AF$ and hence find the area of triangle $AFB$.</span><span class="inline-marks">[5]</span></div></li></ol>

The planes $p_1$, $p_2$ and $p_3$ have equations \[p_1 : -x + 2y + 4z = 5,\]\[p_2 : 3x + ay - 6z = 6,\]\[p_3 : 2x - 4y - 8z = b,\]where $a$ and $b$ are real constants, $b > 0$.

The line $l$ has equation $\frac{x - 5}{8} = \frac{3 - y}{6} = \frac{z - 1}{5}$. Given that the planes $p_1$ and $p_2$ intersect at $l$, show that $a = -1$.

Find the acute angle between $p_1$ and $p_2$.

The plane $p_4$ has equation $\mathbf{r} \cdot \begin{pmatrix} 1 \\ -2 \\ -4 \end{pmatrix} = 0$. Given that $p_1$ and $p_3$ are equidistant from $p_4$, find the value of $b$.

The point $A$ with coordinates $(-1, 0, 1)$ lies on $p_1$, and $F$ is the foot of the perpendicular from $A$ to $l$. The point $B$ is the reflection of $F$ in the plane $p_3$. Find the length of $AF$ and hence find the area of triangle $AFB$.

2018 HCI CT Q8

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzU5NjI5NTM2NzEtbDVrb2J0c3JtNW4uanBnIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTc3MywiZXhwIjoxNzkyNjk2MDczfQ.OLhNYy-JCGRA8Ye-VnCJ-hUjzQcv6ZlZyPehBGOWwos" 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>A line $l$ passes through the points $A$ and $B$ with coordinates $(0, -2, 2)$ and $(1, 0, 1)$ respectively.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the acute angle between $\overrightarrow{OA}$ and the line $l$, where&nbsp;$O$ is the origin.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, find the shortest distance from $O$ to the line $l$, leaving your answer in exact form.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>A plane $\pi_1$ has equation $\mathbf{r} \cdot (2\mathbf{i} - \mathbf{j}) = 2$.</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Show that the line $l$ lies in the plane $\pi_1$.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>A second plane $\pi_2$ contains the line $l$ and is perpendicular to the plane $\pi_1$.</p>
<ol class="i-list" start="4"><li><div class="list-item-content"><span class="list-item-text">Find the cartesian equation of the plane $\pi_2$.</span><span class="inline-marks">[2]</span></div></li></ol>

A line $l$ passes through the points $A$ and $B$ with coordinates $(0, -2, 2)$ and $(1, 0, 1)$ respectively.

Find the acute angle between $\overrightarrow{OA}$ and the line $l$, where $O$ is the origin.

Hence, find the shortest distance from $O$ to the line $l$, leaving your answer in exact form.

A plane $\pi_1$ has equation $\mathbf{r} \cdot (2\mathbf{i} - \mathbf{j}) = 2$.

Show that the line $l$ lies in the plane $\pi_1$.

A second plane $\pi_2$ contains the line $l$ and is perpendicular to the plane $\pi_1$.

Find the cartesian equation of the plane $\pi_2$.

2015 MI P1 Q7

<p>In a computer game, a slope can be modelled as a plane $p$ containing three points, $A(1, 0, -3)$, $B(1, 4, -15)$ and $C(2, 3, -14)$.</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&nbsp;$p$ has equation $\mathbf{r} \cdot \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix} = -1$.<span class="inline-marks">[2]</span></div></li></ol>
<p>In Stage $1$ of the game, Griffles travels on the slope from $A$ along a path with equation $\mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ -3 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ m \\ n \end{pmatrix}$ where&nbsp;$m$ and $n$ are constants and $\lambda$ is a real parameter.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find an equation relating $m$ and $n$.<span class="inline-marks">[1]</span></div></li></ol>
<p>In the rest of the question, it is given that&nbsp;$m = \frac{1}{3}$, $n = -3$ and Griffles is modelled as a point travelling on the slope. An obstacle in the form of a rectangular virtual screen stands on the slope with its base modelled by the line segment $BC$ (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/1769231053046-qezc46gihlc.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkyMzEwNTMwNDYtcWV6YzQ2Z2lobGMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTc3MywiZXhwIjoxNzkyNjk2MDczfQ.3M2ydzaQwzaC4ladiwoRX1w8jLR6VUXFWfUTQsiVRjc" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list" start="3"><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph"><p>By considering the <strong>line segment</strong> $BC$, show that Griffles is able to successfully navigate this obstacle without colliding into it.</p><span class="inline-marks">[3]</span></div></li></ol>
<p>A laser gun with a sensor is mounted at the point $E(5, 2, 1)$ above the slope. The sensor is activated when an object is within a range of $5$ units.</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Show that Griffles does not activate the sensor.<span class="inline-marks">[3]</span></div></li></ol>
<p>After completing Stage $1$, Griffles is teleported to another slope. This slope can be modelled as a plane $q$ parallel to $p$ such that&nbsp;$E$ is equidistant from $p$ and $q$.</p>
<ol class="abc-list" start="5"><li data-list-item-id="item-e" data-list-item-path="item-e"><div class="list-item-paragraph">Find the cartesian equation of $q$.<span class="inline-marks">[3]</span></div></li></ol>

In a computer game, a slope can be modelled as a plane $p$ containing three points, $A(1, 0, -3)$, $B(1, 4, -15)$ and $C(2, 3, -14)$.

In Stage $1$ of the game, Griffles travels on the slope from $A$ along a path with equation $\mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ -3 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ m \\ n \end{pmatrix}$ where $m$ and $n$ are constants and $\lambda$ is a real parameter.

Find an equation relating $m$ and $n$.

In the rest of the question, it is given that $m = \frac{1}{3}$, $n = -3$ and Griffles is modelled as a point travelling on the slope. An obstacle in the form of a rectangular virtual screen stands on the slope with its base modelled by the line segment $BC$ (see diagram).

A laser gun with a sensor is mounted at the point $E(5, 2, 1)$ above the slope. The sensor is activated when an object is within a range of $5$ units.

Show that Griffles does not activate the sensor.

After completing Stage $1$, Griffles is teleported to another slope. This slope can be modelled as a plane $q$ parallel to $p$ such that $E$ is equidistant from $p$ and $q$.

2025 RI P2 Q4

<p>Referred to the origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where&nbsp;$\mathbf{a}$ and $\mathbf{b}$ are non-zero and non-parallel vectors. The point $C$ lies on $OA$ such that&nbsp;$OC : CA = 1 : 2$. The point $D$ lies on $OB$ such that&nbsp;$OB : DB = 3 : 1$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the position vectors $\overrightarrow{OC}$ and $\overrightarrow{OD}$, giving your answers in terms of $\mathbf{a}$ and $\mathbf{b}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that the point $E$ where the lines $AD$ and $BC$ meet has position vector \[\frac{1}{7}\mathbf{a} + \frac{4}{7}\mathbf{b}\]</span><span class="inline-marks">[5]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that the area of triangle $OAE$ can be written as&nbsp;$k|\mathbf{a} \times \mathbf{b}|$, where&nbsp;$k$ is a constant to be found.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>It is further given that&nbsp;$\mathbf{b}$ is a unit vector.</p>
<ol class="i-list" start="4"><li><div class="list-item-content"><span class="list-item-text">Give the geometrical meaning of $|\mathbf{a} \cdot \mathbf{b}|$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">If&nbsp;$|\mathbf{a} \cdot \mathbf{b}| = \frac{1}{2}|\mathbf{a}|$, find the perpendicular distance of point $A$ to $OB$, leaving your answer in terms of $|\mathbf{a}|$.</span><span class="inline-marks">[3]</span></div></li></ol>

Referred to the origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a}$ and $\mathbf{b}$ are non-zero and non-parallel vectors. The point $C$ lies on $OA$ such that $OC : CA = 1 : 2$. The point $D$ lies on $OB$ such that $OB : DB = 3 : 1$.

Show that the point $E$ where the lines $AD$ and $BC$ meet has position vector \[\frac{1}{7}\mathbf{a} + \frac{4}{7}\mathbf{b}\]

Show that the area of triangle $OAE$ can be written as $k|\mathbf{a} \times \mathbf{b}|$, where $k$ is a constant to be found.

It is further given that $\mathbf{b}$ is a unit vector.

Give the geometrical meaning of $|\mathbf{a} \cdot \mathbf{b}|$.

If $|\mathbf{a} \cdot \mathbf{b}| = \frac{1}{2}|\mathbf{a}|$, find the perpendicular distance of point $A$ to $OB$, leaving your answer in terms of $|\mathbf{a}|$.

Projection Lengths

2025 MI P1 Q11

<p>With reference to the origin $O$, the points $A$, $P$ and $Q$ have position vectors $\mathbf{a}$, $\mathbf{p}$ and $\mathbf{q}$ respectively. A straight line $l$ through the point $A$ is parallel to a unit vector $\mathbf{e}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">The point $Q$ lies on $l$. Show that&nbsp;$(\mathbf{q} - \mathbf{a}) \times \mathbf{e} = \mathbf{0}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$P$ is not on $l$, give the geometrical meaning of $|(\mathbf{p} - \mathbf{a}) \times \mathbf{e}|$.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>It is also given that&nbsp;$\mathbf{p} = 3\mathbf{q}$ and $AQ$ is $2$ units.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Using the results in <strong>(a)</strong> and <strong>(b)</strong>, or otherwise, show that the area of triangle $APQ$ can be written as&nbsp;$k|\mathbf{q} \times \mathbf{e}|$, where&nbsp;$k$ is a constant to be determined.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, or otherwise, find the acute angle between $l$ and $PQ$, given further that the area of triangle $\Delta APQ$ is $3$ units${}^2$ and $|\mathbf{q}| = 2$.</span><span class="inline-marks">[3]</span></div></li></ol>

With reference to the origin $O$, the points $A$, $P$ and $Q$ have position vectors $\mathbf{a}$, $\mathbf{p}$ and $\mathbf{q}$ respectively. A straight line $l$ through the point $A$ is parallel to a unit vector $\mathbf{e}$.

Given that $P$ is not on $l$, give the geometrical meaning of $|(\mathbf{p} - \mathbf{a}) \times \mathbf{e}|$.

It is also given that $\mathbf{p} = 3\mathbf{q}$ and $AQ$ is $2$ units.

Hence, or otherwise, find the acute angle between $l$ and $PQ$, given further that the area of triangle $\Delta APQ$ is $3$ units${}^2$ and $|\mathbf{q}| = 2$.

Parallel Vectors

2025 JPJC P2 Q2

<p>In the diagram $R$ is the point $\left( 0,-4 \right)$ and $P$ is a point on the $y$-axis.<br>The line $PQ$ meets the horizontal line through $R$ at $Q$.</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="i-list"><li><div class="list-item-paragraph">State the equation of $QR$.</div></li><li><div class="list-item-paragraph">Given that the equation of the line $PQ$ is $2y-5x-4=0$, find the coordinates of $P$ and $Q$.</div></li><li><div class="list-item-paragraph">Given that the length of $PQ$ is $\sqrt{t}$, find the value of $t$.</div></li><li><div class="list-item-paragraph">Find the area of triangle $PQR$.</div></li><li><div class="list-item-paragraph">Hence, calculate the perpendicular distance from $R$ to $PQ$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765900124482-mmithszwr5d.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjU5MDAxMjQ0ODItbW1pdGhzendyNWQucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTc3MywiZXhwIjoxNzkyNjk2MDczfQ.-XkFkbjTqCwm13dAtY01k3KMYroWuQNCoEwHjqwzfAU" style="max-width: min(400px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

In the diagram $R$ is the point $\left( 0,-4 \right)$ and $P$ is a point on the $y$-axis.
The line $PQ$ meets the horizontal line through $R$ at $Q$.

Length of a Line Segment

Area of Rectilinear Figures

Equation of a Straight Line

Coordinate Geometry with Line Intersections and Perpendicular Distance

Coordinate Geometry

<p>The diagram shows the graph of $y=8-x$. <br>The graph intersects the $x$-axis and $y$-axis at points $A$ and $B$ respectively.</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="i-list"><li><div class="list-item-paragraph">Write down the coordinates of $A$ and $B$.</div></li><li><div class="list-item-paragraph">Calculate the length of the line segment $AB$.  </div></li><li><div class="list-item-paragraph">Calculate the area of triangle $AOB$.  <br>Hence, calculate the shortest distance from point $O$ to the line $AB$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1765871800920-x8q8ggsxsvk.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjU4NzE4MDA5MjAteDhxOGdnc3hzdmsucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTc3MywiZXhwIjoxNzkyNjk2MDczfQ.yvFBQlClNjij0XKJ4BB2hZwTcXueQvAQzGKd3sIQBTg" style="max-width: min(250px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The diagram shows the graph of $y=8-x$. 
The graph intersects the $x$-axis and $y$-axis at points $A$ and $B$ respectively.

Coordinate Geometry: Line Intercepts and Triangle Area

<p>The triangle $PQR$ has vertices $P\left( -5,11 \right)$, $Q\left( -7,5 \right)$ and $R\left( 10,6 \right)$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Determine whether triangle $PQR$ is a right-angled triangle.</div></li><li><div class="list-item-paragraph">Find the area of triangle $PQR$.</div></li><li><div class="list-item-paragraph">Hence, find the shortest distance from $P$ to $QR$.</div></li></ol>

The triangle $PQR$ has vertices $P\left( -5,11 \right)$, $Q\left( -7,5 \right)$ and $R\left( 10,6 \right)$.

Parallel and Perpendicular Lines

Using the Length to show that a Triangle is Right-angled

<p>The equation of line ${{l}_{1}}$ is given by $\mathbf{r}=\left( \begin{matrix} -12  \\-22  \\11  \\\end{matrix} \right)+\lambda \left( \begin{matrix}  4  \\ 4  \\  -7  \\\end{matrix} \right);\lambda \in \mathbb{R}$ and the point $A$ has coordinates $\left( 3,1,-2 \right)$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Exact shortest distance of $A$ to ${{l}_{1}}$,</div></li><li><div class="list-item-paragraph">Foot of perpendicular of $A$ to ${{l}_{1}}$,</div></li><li><div class="list-item-paragraph">Mirror image of point $A$ in the line ${{l}_{1}}$.</div></li></ol>

The equation of line ${{l}_{1}}$ is given by $\mathbf{r}=\left( \begin{matrix} -12  \\-22  \\11  \\\end{matrix} \right)+\lambda \left( \begin{matrix}  4  \\ 4  \\  -7  \\\end{matrix} \right);\lambda \in \mathbb{R}$ and the point $A$ has coordinates $\left( 3,1,-2 \right)$.

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