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PSLE

PSLE Foundation Mathematics

<p>A plane $\pi_1$ contains two vectors $\begin{pmatrix}2\\1\\0\end{pmatrix}$ and $\begin{pmatrix}0\\1\\2\end{pmatrix}$, and passes through the point $A$ with coordinates $(1,0,1)$.</p>
<ol class="abc-list"><li data-list-item-id="tjc-q08-item01" data-list-item-path="tjc-q08-item01"><div class="list-item-paragraph">Show that the cartesian equation of $\pi_1$ is $x-2y+z=2$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="tjc-q08-item02" data-list-item-path="tjc-q08-item02"><div class="list-item-paragraph">The line $l$ contains the point $B$ with coordinates $(2,1,0)$ and is parallel to the vector $\begin{pmatrix}1\\2\\k\end{pmatrix}$. Given that the acute angle between $l$ and $\pi_1$ is $\theta$, where&nbsp;$\sin\theta=\frac13$, find the possible values of $k$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="tjc-q08-item03" data-list-item-path="tjc-q08-item03"><div class="list-item-paragraph">Given now that&nbsp;$k=1$, find the coordinates of the point of intersection between $l$ and $\pi_1$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="tjc-q08-item04" data-list-item-path="tjc-q08-item04"><div class="list-item-paragraph">A plane $\pi_2$ contains the point $C$ with coordinates $(2,1,-3)$ and is parallel to $\pi_1$. Find the exact distance between $\pi_1$ and $\pi_2$.<span class="inline-marks">[3]</span></div></li></ol>

A plane $\pi_1$ contains two vectors $\begin{pmatrix}2\\1\\0\end{pmatrix}$ and $\begin{pmatrix}0\\1\\2\end{pmatrix}$, and passes through the point $A$ with coordinates $(1,0,1)$.

Show that the cartesian equation of $\pi_1$ is $x-2y+z=2$.

The line $l$ contains the point $B$ with coordinates $(2,1,0)$ and is parallel to the vector $\begin{pmatrix}1\\2\\k\end{pmatrix}$. Given that the acute angle between $l$ and $\pi_1$ is $\theta$, where $\sin\theta=\frac13$, find the possible values of $k$.

Given now that $k=1$, find the coordinates of the point of intersection between $l$ and $\pi_1$.

A plane $\pi_2$ contains the point $C$ with coordinates $(2,1,-3)$ and is parallel to $\pi_1$. Find the exact distance between $\pi_1$ and $\pi_2$.

Distance from a Point to a Plane

Angle Between a Line and a Plane

Equations of Planes

Intersection of a Line and a Plane

2026 TJC P1 Q8

Vectors

<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

Line of Intersection of Two Planes

2026 HCI P1 Q8

<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.

Equations of Lines

2025 RI Promo Q10

<p>The line $l$ passes through $A(1, 2, -1)$ and is parallel to $3\mathbf{i} + 3\mathbf{k}$.</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">State the vector equation of $l$.<span class="inline-marks">[1]</span></div></li></ol>
<p>The plane $p$ has equation</p>
<p style="text-align: center">\[\mathbf{r} \cdot \begin{pmatrix} 1 \\ -2 \\ 2 \end{pmatrix} = 13.\]</p>
<ol class="i-list" start="2"><li data-list-item-id="item-ii"><div class="list-item-paragraph">Find the acute angle between $l$ and $p$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iii"><div class="list-item-paragraph">Find the point of intersection of $l$ and $p$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iv"><div class="list-item-paragraph">Find the vector equation of the reflection of $l$ in $p$.<span class="inline-marks">[6]</span></div></li></ol>

The line $l$ passes through $A(1, 2, -1)$ and is parallel to $3\mathbf{i} + 3\mathbf{k}$.

\[\mathbf{r} \cdot \begin{pmatrix} 1 \\ -2 \\ 2 \end{pmatrix} = 13.\]

<p>The line $l$ passes through $A(3, 1, 4)$ and is parallel to $2\mathbf{i} + \mathbf{j} + \mathbf{k}$.</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">State the vector equation of $l$.<span class="inline-marks">[1]</span></div></li></ol>
<p>The plane $p$ has equation</p>
<p style="text-align: center">\[\mathbf{r} \cdot \begin{pmatrix} 2 \\ -1 \\ -1 \end{pmatrix} = 7.\]</p>
<ol class="i-list" start="2"><li data-list-item-id="item-ii"><div class="list-item-paragraph">Find the acute angle between $l$ and $p$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iii"><div class="list-item-paragraph">Find the point of intersection of $l$ and $p$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iv"><div class="list-item-paragraph">Find the vector equation of the reflection of $l$ in $p$.<span class="inline-marks">[6]</span></div></li></ol>

The line $l$ passes through $A(3, 1, 4)$ and is parallel to $2\mathbf{i} + \mathbf{j} + \mathbf{k}$.

\[\mathbf{r} \cdot \begin{pmatrix} 2 \\ -1 \\ -1 \end{pmatrix} = 7.\]

<p>Physicists are investigating the reflective property of a particular reflective surface. The diagram below shows the set-up of a particular experiment, where a laser emitting device was placed at the point with coordinates $(1, 2, 3)$. A laser beam was emitted in the direction parallel to $\mathbf{i} + \mathbf{k}$. The path of the emitted laser beam and its reflected path make the same angle $\theta$ with the reflective surface. The plane containing these two paths is perpendicular to the reflective surface.</p>
<p>Write down the vector equation of the path of the emitted laser beam.<span class="inline-marks">[1]</span></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/1774500717235-6nupuw71o97.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzQ1MDA3MTcyMzUtNm51cHV3NzFvOTcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDg4MiwiZXhwIjoxNzkzNTA1MTgyfQ.aNKJx40oQ3IDPcURtWsXzdPcBqoQzm-v0O1b1R_RmoQ" style="max-width: min(400px, 100%); height: auto; display: block;" alt="Diagram of laser reflection" class="aligncenter custom-width" /></figure></div>
<p>It is known that the reflective surface has equation $x + y + z = 4$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find $\theta$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that the laser beam meets the reflective surface at the point $(0, 2, 2)$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the vector equation of the path of the reflected laser beam.</span><span class="inline-marks">[5]</span></div></li></ol>

Physicists are investigating the reflective property of a particular reflective surface. The diagram below shows the set-up of a particular experiment, where a laser emitting device was placed at the point with coordinates $(1, 2, 3)$. A laser beam was emitted in the direction parallel to $\mathbf{i} + \mathbf{k}$. The path of the emitted laser beam and its reflected path make the same angle $\theta$ with the reflective surface. The plane containing these two paths is perpendicular to the reflective surface.

Write down the vector equation of the path of the emitted laser beam.

It is known that the reflective surface has equation $x + y + z = 4$.

Show that the laser beam meets the reflective surface at the point $(0, 2, 2)$.

Find the vector equation of the path of the reflected laser beam.

2017 VJC P1 Q11

<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 $p$.

Intersection of Two Lines

N2011 P1 Q11

<div class="graph-block" 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<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 $Q$.

Three-Dimensional Geometry

Angle Between Two Planes

2013 DHS P1 Q12

<p>Planes $p$ and $q$ are perpendicular. Plane $p$ has equation $\mathbf{r} \cdot (\mathbf{i} + 2\mathbf{j} - \mathbf{k}) = 6$. Plane $q$ contains the line $l$ with equation $\mathbf{r} = \mathbf{j} - 12\mathbf{k} + \lambda(4\mathbf{i} + \mathbf{j} + 10\mathbf{k})$ where&nbsp;$\lambda$ is a parameter. The point $A$ has coordinates $(0, 1, -12)$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find a cartesian equation of $q$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find a vector equation of the line $m$, where&nbsp;$p$ and $q$ meet.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>$l$ passes through $p$ at point $B$. Point $C$ is the foot of the perpendicular of $A$ to $p$.</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the position vector of $B$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the position vector of $C$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the exact area of $\Delta ABC$.</span><span class="inline-marks">[2]</span></div></li></ol>

Planes $p$ and $q$ are perpendicular. Plane $p$ has equation $\mathbf{r} \cdot (\mathbf{i} + 2\mathbf{j} - \mathbf{k}) = 6$. Plane $q$ contains the line $l$ with equation $\mathbf{r} = \mathbf{j} - 12\mathbf{k} + \lambda(4\mathbf{i} + \mathbf{j} + 10\mathbf{k})$ where $\lambda$ is a parameter. The point $A$ has coordinates $(0, 1, -12)$.

Find a vector equation of the line $m$, where $p$ and $q$ meet.

$l$ passes through $p$ at point $B$. Point $C$ is the foot of the perpendicular of $A$ to $p$.

Areas of Triangles and Parallelograms Using Vectors

2025 YIJC P1 Q10

<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/1769305060067-nxvlg3y1dx8.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMDUwNjAwNjctbnh2bGczeTFkeDgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDg4MiwiZXhwIjoxNzkzNTA1MTgyfQ.FMZUTFq2CGNIm3XpkZH2aM9SBDVo4KDl5ZRSSa_uGK0" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>At a campsite, an $A$-frame tent is pitched on level ground. Its roof is formed by two slanted polyester sheets lying on two intersecting planes,</p>
<p style="text-align: center">\[\pi_1: x + y + az = 6\]</p>
<p style="text-align: center">\[\pi_2: 2x - y + z = 4\]</p>
<p>where&nbsp;$a$ is a positive real constant. The intersection of $\pi_1$ and $\pi_2$ forms a central ridge in the form of a line $AB$. This ridge is extended with a rope, which is secured to the ground at the point $C$ for stability. The ground is assumed to be the horizontal $xy$-plane.</p>
<ol class="abc-list"><li data-list-item-id="item-11-a" data-list-item-path="item-11-a"><div class="list-item-paragraph">Find a vector parallel to the central ridge, giving your answer in terms of $a$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-11-b" data-list-item-path="item-11-b"><div class="list-item-paragraph">If&nbsp;$C$ lies on $AB$ produced, show that the coordinates of $C$ is $\left(\frac{10}{3}, \frac{8}{3}, 0\right)$.<span class="inline-marks">[3]</span></div><div class="list-item-paragraph">Hence, or otherwise, write down a vector equation of line $AB$ in term of $a$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-11-c" data-list-item-path="item-11-c"><div class="list-item-paragraph">If the angle between the two slanted polyester sheets is $60{}^\circ$, find the value of $a$.<span class="inline-marks">[3]</span></div></li></ol>
<p>It is now given that&nbsp;$a = 1$.</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-11-d" data-list-item-path="item-11-d"><div class="list-item-paragraph">A string is tied between two hooks inside the tent to hang decorations. One hook is fixed at the point $D(2, 3, 1)$ which lies on $\pi_1$. The other hook is located at the point $E$ which lies on $\pi_2$ such that the string $DE$ is perpendicular to $\pi_2$. Find the coordinates of $E$.<span class="inline-marks">[3]</span></div></li></ol>

At a campsite, an $A$-frame tent is pitched on level ground. Its roof is formed by two slanted polyester sheets lying on two intersecting planes,

where $a$ is a positive real constant. The intersection of $\pi_1$ and $\pi_2$ forms a central ridge in the form of a line $AB$. This ridge is extended with a rope, which is secured to the ground at the point $C$ for stability. The ground is assumed to be the horizontal $xy$-plane.

Find a vector parallel to the central ridge, giving your answer in terms of $a$.

If $C$ lies on $AB$ produced, show that the coordinates of $C$ is $\left(\frac{10}{3}, \frac{8}{3}, 0\right)$.

If the angle between the two slanted polyester sheets is $60{}^\circ$, find the value of $a$.

A string is tied between two hooks inside the tent to hang decorations. One hook is fixed at the point $D(2, 3, 1)$ which lies on $\pi_1$. The other hook is located at the point $E$ which lies on $\pi_2$ such that the string $DE$ is perpendicular to $\pi_2$. Find the coordinates of $E$.

2025 TJC P1 Q11

<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>The line ${{l}_{3}}$ which passes through $B\left( 7,4,-5 \right)$ and is parallel to the direction vector $\left( \begin{matrix}4  \\4  \\-7  \\\end{matrix} \right)$. The plane ${{p}_{1}}$ has the equation $x-3y+2z=-5$. Find</p>
<ol class="i-list"><li><div class="list-item-paragraph">the coordinates of point of intersection between the line ${{l}_{3}}$ and plane ${{p}_{1}}$,</div></li><li><div class="list-item-paragraph">the acute angle between the line ${{l}_{3}}$ and plane ${{p}_{1}}$,</div></li><li><div class="list-item-paragraph">the equation of line of the image of line ${{l}_{3}}$ in plane ${{p}_{1}}$.</div></li></ol>

The line ${{l}_{3}}$ which passes through $B\left( 7,4,-5 \right)$ and is parallel to the direction vector $\left( \begin{matrix}4  \\4  \\-7  \\\end{matrix} \right)$. The plane ${{p}_{1}}$ has the equation $x-3y+2z=-5$. Find

Unit 13.2 Worked Example 4

<p>Find the coordinates of point of intersection between the line $l:\frac{x-5}{2}=-(y+2)=z-3$ and the plane $3x+2y-2z=-2$.</p>

Find the coordinates of point of intersection between the line $l:\frac{x-5}{2}=-(y+2)=z-3$ and the plane $3x+2y-2z=-2$.

<p>Find the coordinates of point of intersection between the line $l:\frac{x-2}{2}=y-2=-(z+4)$ and the plane $3x+2y-2z=-2$ .</p>

Find the coordinates of point of intersection between the line $l:\frac{x-2}{2}=y-2=-(z+4)$ and the plane $3x+2y-2z=-2$ .

<p>The line $l$ contains the point $P$ with coordinates $\left( 1,-5,-5 \right)$ and is parallel to $3\mathbf{i}+\mathbf{j}-6\mathbf{k}$. The plane ${{\pi }_{1}}$ has cartesian equation $2x-5y-3z=4$.</p>
<ol class="i-list"><li data-list-item-id="item-1761382001258-53pw380gr" data-list-item-path="item-1761382001258-53pw380gr"><div class="list-item-paragraph">Find the acute angle between $l$ and ${{\pi }_{1}}$, giving your answer to the nearest degree.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1761382001925-ouujl9h4g" data-list-item-path="item-1761382001925-ouujl9h4g"><div class="list-item-paragraph">Find the coordinates of $Q$, the point of intersection between $l$ and ${{\pi }_{1}}$.<span class="inline-marks">[3]</span></div></li></ol>
<p>The plane ${{\pi }_{2}}$ contains $l$ and is perpendicular to ${{\pi }_{1}}$.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1761382022625-kp43rdbw9" data-list-item-path="item-1761382022625-kp43rdbw9"><div class="list-item-paragraph">Find a cartesian equation of ${{\pi }_{2}}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1761382023704-lpg7m2oy5" data-list-item-path="item-1761382023704-lpg7m2oy5"><div class="list-item-paragraph">Find a vector equation of the line where&nbsp;${{\pi }_{1}}$ meets ${{\pi }_{2}}$.<span class="inline-marks">[2]</span></div></li></ol>

The line $l$ contains the point $P$ with coordinates $\left( 1,-5,-5 \right)$ and is parallel to $3\mathbf{i}+\mathbf{j}-6\mathbf{k}$. The plane ${{\pi }_{1}}$ has cartesian equation $2x-5y-3z=4$.

The plane ${{\pi }_{2}}$ contains $l$ and is perpendicular to ${{\pi }_{1}}$.

2024 JPJC Promo Q12

<p>The plane $p$ has equation $\mathbf{r}=\left( \begin{matrix}
   \begin{matrix}
   1  \\
   3  \\
   4  \\
\end{matrix}  \\
\end{matrix} \right)+s\left( \begin{matrix}
   \begin{matrix}
   2  \\
   0  \\
   1  \\
\end{matrix}  \\
\end{matrix} \right)+t\left( \begin{matrix}
   \begin{matrix}
   0  \\
   1  \\
   -1  \\
\end{matrix}  \\
\end{matrix} \right)$, and the line $l$ has the equation 	$\mathbf{r}=\left( \begin{matrix}
   \begin{matrix}
   1  \\
   1  \\
   3  \\
\end{matrix}  \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
   \begin{matrix}
   -1  \\
   2  \\
   2  \\
\end{matrix}  \\
\end{matrix} \right)$, where $\lambda $, $s$ and $t$ are parameters.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that $l$ is perpendicular to $p$ and find the coordinates of the point where $l$ and $\mathrm{p}$ intersect.</div><p style="text-align: right;">[5]</p></li></ol>
<p>The line $m$ meets $l$ at the point $A\left( 1,1,3 \right)$ and it meets $p$ at the point $B\left( 1,3,4 \right)$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">The line $n$ is the reflection of $m$ in $p$. Show that a vector equation of $n$ is $\mathbf{r}=\left( \begin{matrix}
   \begin{matrix}
   1  \\
   3  \\
   4  \\
\end{matrix}  \\
\end{matrix} \right)+\mu \left( \begin{matrix}
   \begin{matrix}
   -4  \\
   2  \\
   5  \\
\end{matrix}  \\
\end{matrix} \right)$, 	$\mu \in \mathbb{R}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the acute angle between $n$ and $p$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Hence or otherwise, find the area bounded by the lines $l$, $m$ and $n$.</div><p style="text-align: right;">[3]</p></li></ol>

The plane $p$ has equation $\mathbf{r}=\left( \begin{matrix}
   \begin{matrix}
   1  \\
   3  \\
   4  \\
\end{matrix}  \\
\end{matrix} \right)+s\left( \begin{matrix}
   \begin{matrix}
   2  \\
   0  \\
   1  \\
\end{matrix}  \\
\end{matrix} \right)+t\left( \begin{matrix}
   \begin{matrix}
   0  \\
   1  \\
   -1  \\
\end{matrix}  \\
\end{matrix} \right)$, and the line $l$ has the equation 	$\mathbf{r}=\left( \begin{matrix}
   \begin{matrix}
   1  \\
   1  \\
   3  \\
\end{matrix}  \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
   \begin{matrix}
   -1  \\
   2  \\
   2  \\
\end{matrix}  \\
\end{matrix} \right)$, where $\lambda $, $s$ and $t$ are parameters.

The line $m$ meets $l$ at the point $A\left( 1,1,3 \right)$ and it meets $p$ at the point $B\left( 1,3,4 \right)$.

2024 CJC P2 Q4

<p>The line ${{l}_{3}}$ which passes through $B(3,-4,4)$ and is parallel to the direction vector $\left( \begin{matrix}14  \\  -2  \\5  \\\end{matrix} \right)$.</p>
<p>The plane ${{p}_{1}}$ has the equation ${{p}_{1}}:3x-2y-3z=4$. Find</p>
<ol class="i-list"><li><div class="list-item-paragraph">the coordinates of point of intersection between the line ${{l}_{3}}$ and plane ${{p}_{1}}$,</div></li><li><div class="list-item-paragraph">the acute angle between the line ${{l}_{3}}$ and plane ${{p}_{1}}$,</div></li><li><div class="list-item-paragraph">the equation of line of the image of line ${{l}_{3}}$ in plane ${{p}_{1}}$.</div></li></ol>

The line ${{l}_{3}}$ which passes through $B(3,-4,4)$ and is parallel to the direction vector $\left( \begin{matrix}14  \\  -2  \\5  \\\end{matrix} \right)$.

The plane ${{p}_{1}}$ has the equation ${{p}_{1}}:3x-2y-3z=4$. Find

<p>The line ${{l}_{3}}$ which passes through $B\left( 7,4,-5 \right)$ and is parallel to the direction vector $\left( \begin{matrix}4  \\4  \\-7  \end{matrix} \right)$. The plane ${{p}_{1}}$ has the equation $x-3y+2z=-5$. Find</p>
<ol class="i-list"><li data-list-item-id="item-1761379971194-hebgcdiog" data-list-item-path="item-1761379971194-hebgcdiog"><div class="list-item-paragraph">the coordinates of point of intersection between the line ${{l}_{3}}$ and plane ${{p}_{1}}$,</div></li><li data-list-item-id="item-1761379971764-2tn38l3l7" data-list-item-path="item-1761379971764-2tn38l3l7"><div class="list-item-paragraph">the acute angle between the line ${{l}_{3}}$ and plane ${{p}_{1}}$,</div></li><li data-list-item-id="item-1761379972386-5ppz5x9px" data-list-item-path="item-1761379972386-5ppz5x9px"><div class="list-item-paragraph">the equation of line of the image of line ${{l}_{3}}$ in plane ${{p}_{1}}$.</div></li></ol>

The line ${{l}_{3}}$ which passes through $B\left( 7,4,-5 \right)$ and is parallel to the direction vector $\left( \begin{matrix}4  \\4  \\-7  \\\end{matrix} \right)$. The plane ${{p}_{1}}$ has the equation $x-3y+2z=-5$. Find

<p></p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: left; --cell-justify: flex-start"><h3><strong>Relationships Between</strong></h3></th><th style="text-align: left; --cell-justify: flex-start"><h3>Essential Skills Checklist</h3></th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start"><h3>A point and a line</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Shortest distance of the point to the line</p><p>☐ Foot of perpendicular</p><p>☐ Mirror image of a point in a line</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>A line segment and a line</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Length of projection of a line segment on a line</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>$2$ parallel lines</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ $2$ parallel lines</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>$2$ Intersecting Lines</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Point of intersection</p><p>☐ Angle between $2$ intersecting lines</p><p>☐ Mirror image of a line in another line</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>$2$ skew Lines</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Angle between them</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>A point and a plane</h3><h3></h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Deduce a random point on plane</p><p>☐ Shortest distance of point to plane</p><p>☐ Foot of perpendicular on the plane</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>The origin and a plane</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Equations of planes which contains the origin</p><p>☐ $x-y$, $y-z$ and $x-z$ planes</p><p>☐ Shortest distance of origin to the plane</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>Line segment and a plane</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Length of projection of line segment to the plane</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>Line and a plane</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Point of intersection</p><p>☐ Angle between the line and plane</p><p>☐ Mirror Image of line in plane</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>$2$ Intersecting Planes</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Equation of intersecting line without using GC</p><p>☐ Equation of intersecting line by using GC</p><p>☐ Angle between the planes</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>$2$ Parallel Planes</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Distance between $2$ parallel planes</p><p>☐ Finding equations of planes a fixed distance away from another parallel plane</p></td></tr></tbody></table></div></div>

Perpendicular Projection and Reflection in a Line

Projection Lengths

Finding Points on a Plane

Angles Between Two Lines

Distance from a Point to a Line

<p>The plane ${{\Pi }_{1}}$ and the line $~l$ have equations</p>
<p style="text-align: center">${{\Pi }_{1}}:\mathbf{r}\cdot \left( \begin{matrix}
2 \\
1 \\
-1 \\
\end{matrix} \right)=3$ and $l:\frac{x+2}{3}=\frac{4-y}{2}=z-3$</p>
<p>respectively.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the acute angle between ${{\Pi }_{1}}$ and $l$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the coordinates of the point of intersection between ${{\Pi }_{1}}$ and $l$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the perpendicular distance from $B\left( 10,\,-4,\,7 \right)$ to ${{\Pi }_{1}}$.<span class="inline-marks">[3]</span></div></li></ol>
<p>The plane ${{\Pi }_{2}}$ contains $l$ and is perpendicular to ${{\Pi }_{1}}$.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find a cartesian equation of ${{\Pi }_{2}}$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Without using a calculator, find a vector equation of the line which lies in both&nbsp;${{\Pi }_{1}}$ and ${{\Pi }_{2}}$.<span class="inline-marks">[2]</span></div></li></ol>

The plane ${{\Pi }_{1}}$ and the line $~l$ have equations

${{\Pi }_{1}}:\mathbf{r}\cdot \left( \begin{matrix}
2 \\
1 \\
-1 \\
\end{matrix} \right)=3$ and $l:\frac{x+2}{3}=\frac{4-y}{2}=z-3$

The plane ${{\Pi }_{2}}$ contains $l$ and is perpendicular to ${{\Pi }_{1}}$.

Math Question Bank