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PSLE

PSLE Foundation Mathematics

<p>The plane $\Pi_1$ is defined by the equation $\mathbf{r}\cdot((\beta-1)\mathbf{i}+\beta\mathbf{j}+2\beta\mathbf{k})=-3$, where&nbsp;$\beta$ is a real constant. A point $P$ has coordinates $(3,1,-2)$.</p>
<ol class="abc-list"><li data-list-item-id="hci-q08-item01" data-list-item-path="hci-q08-item01"><div class="list-item-paragraph">Verify that the point $P$ lies on $\Pi_1$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="hci-q08-item02" data-list-item-path="hci-q08-item02"><div class="list-item-paragraph">The plane $\Pi_2$ contains $P$ and is parallel to the vectors $\begin{pmatrix}1\\1\\2\end{pmatrix}$ and $\begin{pmatrix}1\\3\\4\end{pmatrix}$.</div><div class="list-item-paragraph">Find, in terms of $\beta$, the vector equation of the line $l$ where&nbsp;$\Pi_1$ and $\Pi_2$ meet.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="hci-q08-item03" data-list-item-path="hci-q08-item03"><div class="list-item-paragraph">The plane $\Pi_3$ has cartesian equation $x+2y=29-4z$.</div><div class="list-item-paragraph">It is known that there is no common point of intersection between $\Pi_1$, $\Pi_2$ and $\Pi_3$. Comment on the relationship between $l$ and $\Pi_3$. Hence find the value of $\beta$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="hci-q08-item04" data-list-item-path="hci-q08-item04"><div class="list-item-paragraph">It is now given that&nbsp;$\beta=3$.</div><div class="list-item-paragraph">Given that a point $A$ lies on $\Pi_3$ and $P$ is the foot of the perpendicular from $A$ to $\Pi_1$, find the coordinates of $A$.<span class="inline-marks">[4]</span></div></li></ol>

The plane $\Pi_1$ is defined by the equation $\mathbf{r}\cdot((\beta-1)\mathbf{i}+\beta\mathbf{j}+2\beta\mathbf{k})=-3$, where $\beta$ is a real constant. A point $P$ has coordinates $(3,1,-2)$.

Verify that the point $P$ lies on $\Pi_1$.

Perpendicular Projection and Reflection in a Plane

Relationships Between a Point and a Plane

Relationships Between a Line and a Plane

Equations of Planes

Intersection of a Line and a Plane

Line of Intersection of Two Planes

2026 HCI P1 Q8

Vectors

<p>With reference to the origin $O$, the point $A$ has position vector $-\mathbf{i}-3\mathbf{j}+3\mathbf{k}$. The plane $\pi$ and the line $l$ have equations \[\mathbf{r}=\begin{pmatrix}4\\1\\1\end{pmatrix}+\lambda\begin{pmatrix}-1\\3\\2\end{pmatrix}+\mu\begin{pmatrix}2\\3\\-4\end{pmatrix}\text{ and}\hspace{0.5em}\mathbf{r}=\begin{pmatrix}2\\1\\1\end{pmatrix}+\beta\begin{pmatrix}3\\t\\-2\end{pmatrix},\] respectively, where&nbsp;$\lambda$, $\mu$ and $\beta$ are parameters and $t$ is a real value.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-ri-promo-q10-ri-q10-item-block-3-item-0" data-list-item-path="h2-2025-ri-promo-q10-ri-q10-item-block-3-item-0"><div class="list-item-paragraph">It is given that&nbsp;$l$ and $\pi$ intersect at a point.</div><ol class="i-list"><li data-list-item-id="h2-2025-ri-promo-q10-ri-q10-item-block-3-item-0-sub-0" data-list-item-path="h2-2025-ri-promo-q10-ri-q10-item-block-3-item-0/h2-2025-ri-promo-q10-ri-q10-item-block-3-item-0-sub-0"><div class="list-item-paragraph">Find the range of values of $t$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-ri-promo-q10-ri-q10-item-block-3-item-0-sub-1" data-list-item-path="h2-2025-ri-promo-q10-ri-q10-item-block-3-item-0/h2-2025-ri-promo-q10-ri-q10-item-block-3-item-0-sub-1"><div class="list-item-paragraph">Find, in terms of $t$, the coordinates of the point of intersection, $B$, of $l$ and $\pi$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="h2-2025-ri-promo-q10-ri-q10-item-block-3-item-1" data-list-item-path="h2-2025-ri-promo-q10-ri-q10-item-block-3-item-1"><div class="list-item-paragraph">It is given that&nbsp;$A$ lies on $l$.</div><ol class="i-list"><li data-list-item-id="h2-2025-ri-promo-q10-ri-q10-item-block-3-item-1-sub-0" data-list-item-path="h2-2025-ri-promo-q10-ri-q10-item-block-3-item-1/h2-2025-ri-promo-q10-ri-q10-item-block-3-item-1-sub-0"><div class="list-item-paragraph">Show that&nbsp;$t=4$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-ri-promo-q10-ri-q10-item-block-3-item-1-sub-1" data-list-item-path="h2-2025-ri-promo-q10-ri-q10-item-block-3-item-1/h2-2025-ri-promo-q10-ri-q10-item-block-3-item-1-sub-1"><div class="list-item-paragraph">Find the position vector of the foot of perpendicular, $F$, of $A$ onto $\pi$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-ri-promo-q10-ri-q10-item-block-3-item-1-sub-2" data-list-item-path="h2-2025-ri-promo-q10-ri-q10-item-block-3-item-1/h2-2025-ri-promo-q10-ri-q10-item-block-3-item-1-sub-2"><div class="list-item-paragraph">Find the cartesian equation of another line $m$ through $A$, that lies on the plane containing $A$, $B$ and $F$, and $m$ makes the same angle with $\pi$ as&nbsp;$l$ makes with $\pi$.<span class="inline-marks">[4]</span></div></li></ol></li></ol>

With reference to the origin $O$, the point $A$ has position vector $-\mathbf{i}-3\mathbf{j}+3\mathbf{k}$. The plane $\pi$ and the line $l$ have equations \[\mathbf{r}=\begin{pmatrix}4\\1\\1\end{pmatrix}+\lambda\begin{pmatrix}-1\\3\\2\end{pmatrix}+\mu\begin{pmatrix}2\\3\\-4\end{pmatrix}\text{ and }\mathbf{r}=\begin{pmatrix}2\\1\\1\end{pmatrix}+\beta\begin{pmatrix}3\\t\\-2\end{pmatrix},\] respectively, where $\lambda$, $\mu$ and $\beta$ are parameters and $t$ is a real value.

It is given that $l$ and $\pi$ intersect at a point.

Angle Between a Line and a Plane

Equations of Lines

2025 RI Promo Q10

<p>The line $l$ passes through the points $A$ and $B$ with coordinates $(-1, 2, 3)$ and $(5, 14, 11)$ respectively. The plane $p$ has equation $2x + 3y - 6z = -7$.</p>
<ol class="i-list"><li data-list-item-id="imported-item-block-1-item-0" data-list-item-path="imported-item-block-1-item-0"><div class="list-item-paragraph">Show that the line $l$ is parallel but not contained on the plane $p$.</div></li><li data-list-item-id="imported-item-block-1-item-1" data-list-item-path="imported-item-block-1-item-1"><div class="list-item-paragraph">Find the distance of the line $l$ from the plane $p$.</div></li><li data-list-item-id="imported-item-block-1-item-2" data-list-item-path="imported-item-block-1-item-2"><div class="list-item-paragraph">Find a cartesian equation of the plane which contains $l$ and is perpendicular to $p$.</div></li></ol>

The line $l$ passes through the points $A$ and $B$ with coordinates $(-1, 2, 3)$ and $(5, 14, 11)$ respectively. The plane $p$ has equation $2x + 3y - 6z = -7$.

Show that the line $l$ is parallel but not contained on the plane $p$.

Find the distance of the line $l$ from the plane $p$.

Find a cartesian equation of the plane which contains $l$ and is perpendicular to $p$.

Distance from a Point to a Plane

<p>The point $A$ has coordinates $(1, -1, 2)$ and the line $l$ has equation $\mathbf{r} = 2\mathbf{i} - 3\mathbf{j} + \lambda(\mathbf{i} - 2\mathbf{j} + 5\mathbf{k})$. The point $A$ and the line $l$ lies in the plane $p_1$.</p>
<ol class="i-list"><li data-list-item-id="imported-item-block-2-item-0" data-list-item-path="imported-item-block-2-item-0"><div class="list-item-paragraph"><p>Find the position vector of the foot of perpendicular from $A$ to the line $l$.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-2-item-1" data-list-item-path="imported-item-block-2-item-1"><div class="list-item-paragraph">Show that the equation of $p_1$ is $2x + y = 1$.<span class="inline-marks">[2]</span></div></li></ol>
<p>The plane $p_2$ with equation $8x + ay + z = 4$ meets the plane $p_1$ at the line $L$ and the point $B(0, 1, 0)$ lies on $L$.</p>
<ol class="i-list" start="3"><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">Show that&nbsp;$L$ is parallel to $\mathbf{i} - 2\mathbf{j} + (2a - 8)\mathbf{k}$.<span class="inline-marks">[1]</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">Show that&nbsp;$a = 4$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-4-item-2" data-list-item-path="imported-item-block-4-item-2"><div class="list-item-paragraph">Find the angle between $l$ and $p_2$.<span class="inline-marks">[2]</span></div></li></ol>
<p>Another point $C$ lies in the plane $p_1$ such that&nbsp;$BC$ is perpendicular to the line $L$.</p>
<ol class="i-list" start="6"><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">Show that&nbsp;$\overrightarrow{BC}$ is parallel to $\mathbf{k}$. Given that the distance of $C$ from $p_2$ is 5 units, find the possible coordinates of $C$.<span class="inline-marks">[4]</span></div></li></ol>

The point $A$ has coordinates $(1, -1, 2)$ and the line $l$ has equation $\mathbf{r} = 2\mathbf{i} - 3\mathbf{j} + \lambda(\mathbf{i} - 2\mathbf{j} + 5\mathbf{k})$. The point $A$ and the line $l$ lies in the plane $p_1$.

Show that the equation of $p_1$ is $2x + y = 1$.

The plane $p_2$ with equation $8x + ay + z = 4$ meets the plane $p_1$ at the line $L$ and the point $B(0, 1, 0)$ lies on $L$.

Show that $L$ is parallel to $\mathbf{i} - 2\mathbf{j} + (2a - 8)\mathbf{k}$.

Find the angle between $l$ and $p_2$.

Another point $C$ lies in the plane $p_1$ such that $BC$ is perpendicular to the line $L$.

Show that $\overrightarrow{BC}$ is parallel to $\mathbf{k}$. Given that the distance of $C$ from $p_2$ is 5 units, find the possible coordinates of $C$.

Perpendicular Projection and Reflection in a Line

Perpendicular Vectors Using the Scalar Product

<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 $q'$.

Angle Between Two Planes

Distance from a Point to a Line

2024 RI Promo Q9

<p>$\mathbf{v}$ and $\mathbf{n}$ are unit vectors in $3$-dimensional space, with $\mathbf{v} \cdot \mathbf{n} \neq 0$.</p>
<p>Let&nbsp;$P$ be a plane with unit normal $\mathbf{n}$. The reflection of $\mathbf{v}$ in plane $P$, $\mathbf{w}$, is given by the formula\[\mathbf{w} = \mathbf{v} - 2(\mathbf{v} \cdot \mathbf{n})\mathbf{n} . \quad (*)\]</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Calculate the vector $\mathbf{w}$ when&nbsp;$\mathbf{v} = \begin{pmatrix} 0 \\ 0 \\ -1 \end{pmatrix}$ and $\mathbf{n} = \frac{1}{5} \begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix}$.<span class="inline-marks">[2]</span></div></li></ol>
<p>When the unit vectors $\mathbf{v}$ and $\mathbf{n}$ vary, the formula $(*)$ will always result in a unit vector for&nbsp;$\mathbf{w}$.</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">Verify that the vector $\mathbf{w}$ calculated in part <strong>(a)</strong> is a unit vector.<span class="inline-marks">[1]</span></div></li></ol>
<p>Before photovoltaic technology allowed for cost-effective direct conversion of sunlight to electrical energy in solar panels, solar power plants relied on arrays of mirrors to concentrate a large area of sunlight into a receiver. In one such plant (see diagram), the receiver is modelled as a single point $C$ with coordinates $(0, 0, 10)$, with mirrors surrounding it. One particular mirror has centre $M$ with coordinates $(-10, -5, 0)$, and its surface is modelled as part of a plane $P$ passing through $M$ with unit normal $\mathbf{n}$.</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/1779451489366-7psqn7e00y7.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Nzk0NTE0ODkzNjYtN3BzcW43ZTAweTcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDkzMywiZXhwIjoxNzkzNTA1MjMzfQ.v4L0wEE89UY_QmOQ1JJUEREIzJE-VulaFoCGxkLdqt4" style="max-width: min(200px, 100%); height: auto; display: block;" alt="Solar plant diagram showing incoming beam of sunlight reflecting off a mirror at point M towards a receiver at point C." class="aligncenter custom-width" /></figure><div class="image-caption" style="text-align: center;">Solar plant diagram showing incoming beam of sunlight reflecting off a mirror at point M towards a receiver at point C.</div></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">Write down the unit vector $\mathbf{w}$ in the direction of $\overline{MC}$.<span class="inline-marks">[1]</span></div></li></ol>
<p>At midday, incoming beams of sunlight can be modelled as travelling along the direction $\mathbf{v} = \begin{pmatrix} 0 \\ 0 \\ -1 \end{pmatrix}$. The mirror is angled so that a beam of sunlight shining on $M$ is reflected to pass through the receiver $C$.</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">Using the formula $(*)$ or otherwise, derive that a unit normal for plane $P$ is $\mathbf{n} = \frac{1}{\sqrt{30}} \begin{pmatrix} 2 \\ 1 \\ 5 \end{pmatrix}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-e" data-list-item-path="item-e"><div class="list-item-paragraph">Hence find a cartesian equation of plane $P$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-f" data-list-item-path="item-f"><div class="list-item-paragraph">Find the acute angle that plane $P$ makes with the $xy$-plane.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-g" data-list-item-path="item-g"><div class="list-item-paragraph">State, in the context of the question, an interpretation of the condition $\mathbf{v} \cdot \mathbf{n} \neq 0$.<span class="inline-marks">[1]</span></div></li></ol>

$\mathbf{v}$ and $\mathbf{n}$ are unit vectors in $3$-dimensional space, with $\mathbf{v} \cdot \mathbf{n} \neq 0$.

Let $P$ be a plane with unit normal $\mathbf{n}$. The reflection of $\mathbf{v}$ in plane $P$, $\mathbf{w}$, is given by the formula\[\mathbf{w} = \mathbf{v} - 2(\mathbf{v} \cdot \mathbf{n})\mathbf{n} . \quad (*)\]

When the unit vectors $\mathbf{v}$ and $\mathbf{n}$ vary, the formula $(*)$ will always result in a unit vector for $\mathbf{w}$.

Before photovoltaic technology allowed for cost-effective direct conversion of sunlight to electrical energy in solar panels, solar power plants relied on arrays of mirrors to concentrate a large area of sunlight into a receiver. In one such plant (see diagram), the receiver is modelled as a single point $C$ with coordinates $(0, 0, 10)$, with mirrors surrounding it. One particular mirror has centre $M$ with coordinates $(-10, -5, 0)$, and its surface is modelled as part of a plane $P$ passing through $M$ with unit normal $\mathbf{n}$.

At midday, incoming beams of sunlight can be modelled as travelling along the direction $\mathbf{v} = \begin{pmatrix} 0 \\ 0 \\ -1 \end{pmatrix}$. The mirror is angled so that a beam of sunlight shining on $M$ is reflected to pass through the receiver $C$.

Unit Vectors

2024 EJC Promo Q13

<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>
<div class="graph-block" data-graph-config="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" data-width="600" data-height="300" style="width:600px;height:300px;max-width:100%;display:block;margin:0 auto;"></div>
<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 $1$.

Intersection of Two Lines

2023 ACJC P1 Q12

<p>The plane $p$ passes through the points with coordinates $(4, -1, -3)$, $(-2, -5, 2)$ and $(4, -3, -2)$.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find a Cartesian equation of $p$.<span class="inline-marks">[4]</span></div></li></ol>
<p>The line $l_1$ has equation $\frac{x-1}{2} = \frac{y-2}{-4} = \frac{z+3}{1}$ and the line $l_2$ has equation $\frac{x+2}{1} = \frac{y-1}{5} = \frac{z-3}{k}$, where&nbsp;$k$ is a constant. It is given that&nbsp;$l_1$ and $l_2$ intersect.</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 value of $k$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Show that&nbsp;$l_1$ lies in $p$ and find the coordinates of the point at which $l_2$ intersects $p$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">Find the acute angle between $l_2$ and $p$.<span class="inline-marks">[3]</span></div></li></ol>

The plane $p$ passes through the points with coordinates $(4, -1, -3)$, $(-2, -5, 2)$ and $(4, -3, -2)$.

The line $l_1$ has equation $\frac{x-1}{2} = \frac{y-2}{-4} = \frac{z+3}{1}$ and the line $l_2$ has equation $\frac{x+2}{1} = \frac{y-1}{5} = \frac{z-3}{k}$, where $k$ is a constant. It is given that $l_1$ and $l_2$ intersect.

Show that $l_1$ lies in $p$ and find the coordinates of the point at which $l_2$ intersects $p$.

Find the acute angle between $l_2$ and $p$.

N2011 P1 Q11

<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 $AFB$.

Areas of Triangles and Parallelograms Using Vectors

2018 HCI CT Q8

<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 $\pi_2$.

Angles Between Two Lines

2015 MI P1 Q7

<p>Describe the relationship between the plane $\mathbf{r}\cdot (\mathbf{i}+2\mathbf{j}-\mathbf{k})=5$ to the line $x-2=\frac{y}{2}=-(z+1)$. Find the angle between the line and plane. </p>

Describe the relationship between the plane $\mathbf{r}\cdot (\mathbf{i}+2\mathbf{j}-\mathbf{k})=5$ to the line $x-2=\frac{y}{2}=-(z+1)$. Find the angle between the line and plane. 

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkyMzEwNTMwNDYtcWV6YzQ2Z2lobGMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDkzMywiZXhwIjoxNzkzNTA1MjMzfQ.K4VCRpi8Pp3RWpZBkcRb2EiYMUjf95EqyE9sGSomrK4" 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>Relative to the origin $O$, point $A$ has position vector given by $\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$. The line $l_1$ passes through the point $A$ and is parallel to $\begin{pmatrix} t \\ 3 - t^2 \\ 1 \end{pmatrix}$. The plane $p$ has equation $\mathbf{r} = \lambda \begin{pmatrix} 0 \\ 5 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 5 \\ 0 \\ t \end{pmatrix}, \lambda, \mu \in \mathbb{R}$ where&nbsp;$t$ is a real constant. It is known that&nbsp;$l_1$ and $p$ are parallel and $l_1$ is not on $p$.</p>
<ol class="abc-list"><li data-list-item-id="item-1768711344529-8nihmues6" data-list-item-path="item-1768711344529-8nihmues6"><div class="list-item-paragraph">Show that&nbsp;$t = -1$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1768711345279-s547r63ey" data-list-item-path="item-1768711345279-s547r63ey"><div class="list-item-paragraph">The line $l_2$ passes through the origin and point $A$. Find the acute angle between $l_1$ and $l_2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1768711345923-79fmat0do" data-list-item-path="item-1768711345923-79fmat0do"><div class="list-item-paragraph">The vector $\mathbf{n}$ is a unit vector normal to $p$. State the geometrical meaning of $|\mathbf{a} \cdot \mathbf{n}|$ and find the exact value of $|\mathbf{a} \cdot \mathbf{n}|$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1768711346929-qm8zctapw" data-list-item-path="item-1768711346929-qm8zctapw"><div class="list-item-paragraph">Find a vector equation of the line of reflection of $l_2$ in $p$.<span class="inline-marks">[4]</span></div></li></ol>

Relative to the origin $O$, point $A$ has position vector given by $\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$. The line $l_1$ passes through the point $A$ and is parallel to $\begin{pmatrix} t \\ 3 - t^2 \\ 1 \end{pmatrix}$. The plane $p$ has equation $\mathbf{r} = \lambda \begin{pmatrix} 0 \\ 5 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 5 \\ 0 \\ t \end{pmatrix}, \lambda, \mu \in \mathbb{R}$ where $t$ is a real constant. It is known that $l_1$ and $p$ are parallel and $l_1$ is not on $p$.

Projection Lengths

2025 EJC P2 Q4

<p>The line $\ell$ has the equation $\frac{x-a}{2}=y-2=\frac{1-2z}{b}$, where&nbsp;$a$ and $b$ are real constants. The plane $\pi$ has the equation $3x-y+4z=10$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Given that the line $\ell$ and the plane $\pi$ do not intersect, show that&nbsp;$a \neq \frac{10}{3}$ and $b = \frac{5}{2}$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">It is given that&nbsp;$a=1$ and $b=3$. Find the point of intersection between the line $\ell$ and the plane $\pi$.<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">It is given now that&nbsp;$a=1, b=\frac{5}{2}$.</div><ol class="i-list"><li data-list-item-id="item-c-i" data-list-item-path="item-c/item-c-i"><div class="list-item-paragraph">Find the distance between the line $\ell$ and the plane $\pi$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c-ii" data-list-item-path="item-c/item-c-ii"><div class="list-item-paragraph">Determine whether the line $\ell$ and the origin $O$ lie on the same side of the plane $\pi$. State an equation of the other plane that is equidistant from line $\ell$ and parallel to plane $\pi$.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

The line $\ell$ has the equation $\frac{x-a}{2}=y-2=\frac{1-2z}{b}$, where $a$ and $b$ are real constants. The plane $\pi$ has the equation $3x-y+4z=10$.

Given that the line $\ell$ and the plane $\pi$ do not intersect, show that $a \neq \frac{10}{3}$ and $b = \frac{5}{2}$.

It is given that $a=1$ and $b=3$. Find the point of intersection between the line $\ell$ and the plane $\pi$.

It is given now that $a=1, b=\frac{5}{2}$.

2025 ASRJC P1 Q9

<p>Describe the relationship between the plane $\mathbf{r}\cdot(2\mathbf{i}+3\mathbf{j}-5\mathbf{k})=89$ to the line $-\frac{x-4}{2}=-\frac{y}{3}=\frac{z+1}{5}$. Find the angle between the line and plane.</p>

Describe the relationship between the plane $\mathbf{r}\cdot(2\mathbf{i}+3\mathbf{j}-5\mathbf{k})=89$ to the line $-\frac{x-4}{2}=-\frac{y}{3}=\frac{z+1}{5}$. Find the angle between the line and plane.

<p>Describe the relationship between the plane $\mathbf{r}\cdot(4\mathbf{j}-\mathbf{k})=-5$ to the line $2-x=y+1=\frac{z-1}{9}$. Find the angle between the line and plane.</p>

Describe the relationship between the plane $\mathbf{r}\cdot(4\mathbf{j}-\mathbf{k})=-5$ to the line $2-x=y+1=\frac{z-1}{9}$. Find the angle between the line and plane.

<p>Describe the relationship between the plane $\mathbf{r}\cdot(3\mathbf{i}-2\mathbf{j}+4\mathbf{k})=2$ to the line $\frac{1-y}{2}=z,x=2$. Find the angle between the line and plane.</p>

Describe the relationship between the plane $\mathbf{r}\cdot(3\mathbf{i}-2\mathbf{j}+4\mathbf{k})=2$ to the line $\frac{1-y}{2}=z,x=2$. Find the angle between the line and plane.

<p>Describe the relationship between the plane $\mathbf{r}\cdot(2\mathbf{i}+\mathbf{k})=6$ to the line $2-x=\frac{z}{2},y=1$. Find the angle between the line and plane.</p>

Describe the relationship between the plane $\mathbf{r}\cdot(2\mathbf{i}+\mathbf{k})=6$ to the line $2-x=\frac{z}{2},y=1$. Find the angle between the line and plane.

<p>Describe the relationship between the plane $\mathbf{r}\cdot(2\mathbf{i}+\mathbf{k})=4$ to the line $2-x=\frac{z}{2},y=1$. Find the angle between the line and plane.</p>

Describe the relationship between the plane $\mathbf{r}\cdot(2\mathbf{i}+\mathbf{k})=4$ to the line $2-x=\frac{z}{2},y=1$. Find the angle between the line and plane.

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzODE1ODY4NzItMnp6aGx0bG1meGMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDkzMywiZXhwIjoxNzkzNTA1MjMzfQ.NXSX8i42yQh8yTSkaC1uot-GVaMWan1l0UurZdA06vA" 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.

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