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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

<ol class="abc-list"><li data-list-item-id="h2-2025-yijc-promo-q8-yijc-q8-item-block-0-item-0" data-list-item-path="h2-2025-yijc-promo-q8-yijc-q8-item-block-0-item-0"><div class="list-item-paragraph">With reference to the origin $O$, points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. Point $P$ lies on $AB$, such that&nbsp;$AP:PB=2:3$. Given that&nbsp;$OP$ and $AB$ are perpendicular, show that&nbsp;$\mathbf{a}\cdot\mathbf{b}=3|\mathbf{a}|^2-2|\mathbf{b}|^2$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-yijc-promo-q8-yijc-q8-item-block-0-item-1" data-list-item-path="h2-2025-yijc-promo-q8-yijc-q8-item-block-0-item-1"><div class="list-item-paragraph">The plane $\pi$ has equation $\mathbf{r}=\begin{pmatrix}0\\2\\1\end{pmatrix}+\lambda\begin{pmatrix}-1\\2\\3\end{pmatrix}+\mu\begin{pmatrix}3\\0\\-1\end{pmatrix},\ \lambda,\mu\in\mathbb{R}.$</div><ol class="i-list"><li data-list-item-id="item-1789031430179-n1av0zoyd" data-list-item-path="item-1789031430179-n1av0zoyd"><div class="list-item-paragraph"><p>Show that a cartesian equation of $\pi$ is $x-4y+3z=-5$.</p><span class="inline-marks">[3]</span></div></li></ol><div class="list-item-paragraph"><p>The line $l_1$ has equation $\mathbf{r}=\begin{pmatrix}6\\-1\\4\end{pmatrix}+t\begin{pmatrix}1\\1\\1\end{pmatrix},\ t\in\mathbb{R}.$</p></div><ol class="i-list" start="2"><li data-list-item-id="h2-2025-yijc-promo-q8-yijc-q8-item-block-0-item-1-sub-4" data-list-item-path="h2-2025-yijc-promo-q8-yijc-q8-item-block-0-item-1/h2-2025-yijc-promo-q8-yijc-q8-item-block-0-item-1-sub-4"><div class="list-item-paragraph"><p>Find the acute angle between $l_1$ and $\pi$.</p><span class="inline-marks">[3]</span></div></li></ol><div class="list-item-after-content"><p>The line $l_2$ passes through the point $Q(-2,-1,2)$ and is parallel to $2\mathbf{i}+3\mathbf{j}+\mathbf{k}$.</p></div><div class="list-item-after-content"><ol class="i-list" start="3"><li data-list-item-id="item-1789031789556-jh2eio6wg" data-list-item-path="item-1789031789556-jh2eio6wg"><div class="list-item-paragraph"><p>Find the position vector of the foot of the perpendicular from $Q$ to $\pi$.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1789031809165-lneq3t01d" data-list-item-path="item-1789031809165-lneq3t01d"><div class="list-item-paragraph"><p>Find the distance between $l_2$ and $\pi$.</p><span class="inline-marks">[2]</span></div></li></ol></div></li></ol>

With reference to the origin $O$, points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. Point $P$ lies on $AB$, such that $AP:PB=2:3$. Given that $OP$ and $AB$ are perpendicular, show that $\mathbf{a}\cdot\mathbf{b}=3|\mathbf{a}|^2-2|\mathbf{b}|^2$.

Distance from a Point to a Plane

Angle Between a Line and a Plane

Perpendicular Vectors Using the Scalar Product

Ratio Theorem and Section Formula

2025 YIJC Promo Q8

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

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

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

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

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

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

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

Equations of Lines

Three-Dimensional Geometry

2025 TJC Promo Q12

<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.

2025 RI Promo Q10

<p>The line $l$ passes through the points $A(2,-1, 3)$ and $B(4, 1, 5)$. The plane $p_1$ has equation $2x-y+2z=0$.</p>
<ol class="i-list"><li data-list-item-id="h2-2025-mi-promo-q10-mi-q10-item-block-1-item-0" data-list-item-path="h2-2025-mi-promo-q10-mi-q10-item-block-1-item-0"><div class="list-item-paragraph">Find the vector equation of the line $l$.<span class="inline-marks">[2]</span></div></li></ol>
<p>The plane $p_2$ contains $l$ and the point $C$ with coordinates $(3, 6, 0)$.</p>
<ol class="i-list" start="2"><li data-list-item-id="h2-2025-mi-promo-q10-mi-q10-item-block-3-item-0" data-list-item-path="h2-2025-mi-promo-q10-mi-q10-item-block-3-item-0"><div class="list-item-paragraph"><p>Show that a vector perpendicular to $p_2$ is $\begin{pmatrix}-5\\2\\3\end{pmatrix}$.</p><p>Hence find the cartesian equation of $p_2$.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-mi-promo-q10-mi-q10-item-block-3-item-1" data-list-item-path="h2-2025-mi-promo-q10-mi-q10-item-block-3-item-1"><div class="list-item-paragraph">Find the vector equation of the line of intersection between plane $p_1$ and plane $p_2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-mi-promo-q10-mi-q10-item-block-3-item-2" data-list-item-path="h2-2025-mi-promo-q10-mi-q10-item-block-3-item-2"><div class="list-item-paragraph">Find the acute angle between the planes $p_1$ and $p_2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-mi-promo-q10-mi-q10-item-block-3-item-3" data-list-item-path="h2-2025-mi-promo-q10-mi-q10-item-block-3-item-3"><div class="list-item-paragraph">Find the position vector of the point $F$, the foot of perpendicular from point $B$ to the plane $p_1$.<span class="inline-marks">[4]</span></div></li></ol>

The line $l$ passes through the points $A(2,-1, 3)$ and $B(4, 1, 5)$. The plane $p_1$ has equation $2x-y+2z=0$.

Find the vector equation of the line $l$.

The plane $p_2$ contains $l$ and the point $C$ with coordinates $(3, 6, 0)$.

Find the vector equation of the line of intersection between plane $p_1$ and plane $p_2$.

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

Find the position vector of the point $F$, the foot of perpendicular from point $B$ to the plane $p_1$.

Angle Between Two Planes

2025 MI PU1 Promo Q10

<p>The plane $p_1$ has equation $x+y+z=3$ while the line $l_1$ passes through the origin and is parallel to $-2\mathbf{i}+3\mathbf{j}-2\mathbf{k}$. It is given that the point $A$ has position vector $\mathbf{i}$.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-asrjc-promo-q10-asrjc-q10-item-block-1-item-0" data-list-item-path="h2-2025-asrjc-promo-q10-asrjc-q10-item-block-1-item-0"><div class="list-item-paragraph">Find the acute angle, in degrees, between $l_1$ and $p_1$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-asrjc-promo-q10-asrjc-q10-item-block-1-item-1" data-list-item-path="h2-2025-asrjc-promo-q10-asrjc-q10-item-block-1-item-1"><div class="list-item-paragraph">Find the exact coordinates of the foot of perpendicular from the point $A$ to $p_1$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="h2-2025-asrjc-promo-q10-asrjc-q10-item-block-1-item-2" data-list-item-path="h2-2025-asrjc-promo-q10-asrjc-q10-item-block-1-item-2"><div class="list-item-paragraph">Point $B$ is a point on the line $l_1$. It is given that the shortest distance from point $B$ to the plane $p_1$ is $\sqrt{27}$ units, find the possible coordinates of $B$.<span class="inline-marks">[3]</span></div></li></ol>
<p>The plane $p_2$ has equation $3x+4y+\alpha z=7+\alpha$, $\alpha \in \mathbb{R}$.</p>
<ol class="abc-list" start="4"><li data-list-item-id="h2-2025-asrjc-promo-q10-asrjc-q10-item-block-3-item-0" data-list-item-path="h2-2025-asrjc-promo-q10-asrjc-q10-item-block-3-item-0"><div class="list-item-paragraph">It is given that the point $C$ with coordinates $(1,1,1)$ lies on $p_1$. Show that point $C$ also lies on $p_2$. Hence find a vector equation of the line $l_2$, the line of intersection between $p_1$ and $p_2$, in terms of $\alpha$.<span class="inline-marks">[3]</span></div></li></ol>

The plane $p_1$ has equation $x+y+z=3$ while the line $l_1$ passes through the origin and is parallel to $-2\mathbf{i}+3\mathbf{j}-2\mathbf{k}$. It is given that the point $A$ has position vector $\mathbf{i}$.

Find the acute angle, in degrees, between $l_1$ and $p_1$.

Find the exact coordinates of the foot of perpendicular from the point $A$ to $p_1$.

Point $B$ is a point on the line $l_1$. It is given that the shortest distance from point $B$ to the plane $p_1$ is $\sqrt{27}$ units, find the possible coordinates of $B$.

The plane $p_2$ has equation $3x+4y+\alpha z=7+\alpha$, $\alpha \in \mathbb{R}$.

It is given that the point $C$ with coordinates $(1,1,1)$ lies on $p_1$. Show that point $C$ also lies on $p_2$. Hence find a vector equation of the line $l_2$, the line of intersection between $p_1$ and $p_2$, in terms of $\alpha$.

2025 ASRJC Promo Q10

<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 $q'$.

Distance from a Point to a Line

2024 RI Promo Q9

<p>Referring to the origin $O$, two planes $\pi_1$ and $\pi_2$ are given by</p>
<p style="text-align: center">\[\pi_1 : \mathbf{r} \cdot \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix} = 18 \quad \text{and} \quad \pi_2 : \mathbf{r} \cdot \begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix} = 3.\]</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">Given that point $A(2, 5, 2)$ lies on $\pi_2$, find $\overrightarrow{OB}$, where&nbsp;$B$ is the image of $A$ when reflected in the plane $\pi_1$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-ii"><div class="list-item-paragraph">Find a vector equation of the line of intersection of $\pi_1$ and $\pi_2$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1781324042384-buo6cn6yi"><div class="list-item-paragraph">Find the Cartesian equation of the plane which is the image of $\pi_2$ when&nbsp;$\pi_2$ is reflected in $\pi_1$.<span class="inline-marks">[3]</span></div></li></ol>

Referring to the origin $O$, two planes $\pi_1$ and $\pi_2$ are given by

\[\pi_1 : \mathbf{r} \cdot \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix} = 18 \quad \text{and} \quad \pi_2 : \mathbf{r} \cdot \begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix} = 3.\]

<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>$\mathbf{v}$ 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Nzk0NTE0ODkzNjYtN3BzcW43ZTAweTcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDg4OCwiZXhwIjoxNzkzNTA1MTg4fQ.QTLm8ZF62-K8AGSjkD3uMtwyTGFXud0bKJ1YrfsgCmU" 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>The line $l_1$ has the vector equation $\mathbf{r} = \begin{pmatrix} a \\ 1 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 0 \\ 1 \\ b \end{pmatrix}$, $\lambda \in \mathbb{R}$, where&nbsp;$a$ and $b$ are real constants.</p>
<p>The plane $p_1$ has cartesian equation $x + 2y + 4z = 4$.</p>
<p>The line $l_1$ is perpendicular to the line $l_2$ with cartesian equation $\frac{1-y}{2} = z - 3$, $x = 5$.</p>
<p>The line $l_1$ intersects the plane $p_1$ at the point with coordinates $(a, 1, 0)$.</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">By first finding the vector equation of line $l_2$, show that&nbsp;$a = b = 2$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-ii"><div class="list-item-paragraph">Verify that the point $A$ with coordinates $(2, 2, 2)$ lies on $l_1$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-iii"><div class="list-item-paragraph">Find the coordinates of the foot of perpendicular, $F$, from point $A$ to the plane $p_1$.<br>Hence find the shortest distance from point $A$ to the plane $p_1$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-iv"><div class="list-item-paragraph">The plane $p_2$ contains the line $l_1$ and the point $F$. Find the vector equation of the plane $p_2$, giving your answer in the form $\mathbf{r} \cdot \mathbf{n} = d$, where&nbsp;$\mathbf{n}$ is a vector and $d$ is a real constant to be determined.<span class="inline-marks">[2]</span></div></li></ol>

The line $l_1$ has the vector equation $\mathbf{r} = \begin{pmatrix} a \\ 1 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 0 \\ 1 \\ b \end{pmatrix}$, $\lambda \in \mathbb{R}$, where $a$ and $b$ are real constants.

The plane $p_1$ has cartesian equation $x + 2y + 4z = 4$.

The line $l_1$ is perpendicular to the line $l_2$ with cartesian equation $\frac{1-y}{2} = z - 3$, $x = 5$.

The line $l_1$ intersects the plane $p_1$ at the point with coordinates $(a, 1, 0)$.

2024 ACJC Promo Q11

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzQ1MDA3MTcyMzUtNm51cHV3NzFvOTcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDg4OCwiZXhwIjoxNzkzNTA1MTg4fQ.N4FfrDrWE9mQ1eM7yvTewM5eBGdvZzJUL4g52vJqaDg" 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 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>To construct a special tentage, $3$ steel poles $OR$, $AQ$ and $BP$ are secured to the horizontal ground such that each of the poles are perpendicular to the ground. The point $O$ on the ground is taken as the origin and the unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OB$ and $OR$ respectively, with units in metres. It is given that&nbsp;$OR=12$, $AQ=9$, $BP=6$, $OA=20$ and $OB=15$ (see diagram).</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1771903221872-mq6u65oef5.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzE5MDMyMjE4NzItbXE2dTY1b2VmNS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0ODg4LCJleHAiOjE3OTM1MDUxODh9.VK0eRgPF2Wc96mz6oROExefMqPIsrg5qrsmvo1Oy_8k" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find $\vec{PR}$ and $\vec{PQ}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph"><p>The canvas roof $PQR$, which is of negligible thickness, is part of the plane $p$. Using your answers in part (i), show that the equation of $p$ is given by ${\mathbf{r} \cdot \begin{pmatrix} 3 \\ 8 \\ 20 \end{pmatrix} = 240.}$</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Find the acute angle between the canvas roof and the plane $OAB$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">A cable is to be installed close to the tentage. This cable is part of the line $L$ with vector equation 	$\mathbf{r}=\left( \begin{matrix}
   \begin{matrix}
   -2  \\
   8  \\
   12  \\
\end{matrix}  \\
\end{matrix} \right)+\lambda \left( \begin{matrix}
   \begin{matrix}
   20  \\
   13  \\
   -4  \\
\end{matrix}  \\
\end{matrix} \right)$, where&nbsp;$\lambda \in \mathbb{R}$. Show that this cable is not parallel to the edge $PR$ of the roof and does not meet 	the edge $PR$. <span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-v" data-list-item-path="item-v"><div class="list-item-paragraph">A lamp is hung vertically downwards from the canvas roof using a cable such that it is at the point $M$ with coordinates 	$\left( \frac{20}{3},5,8 \right)$. To secure the lamp, a second cable is connected from the lamp to the point $N$ on the roof. It is assumed 	that the cables are laid in straight lines and have negligible thickness. For the shortest length to be used for the second 	cable, show that&nbsp;$\overrightarrow{ON}=\left( \begin{matrix}
   \begin{matrix}
   s  \\
   \frac{2525}{473}  \\
   t  \\
\end{matrix}  \\
\end{matrix} \right)$, where&nbsp;$s$ and $t$ are constants to be found. <br><span class="inline-marks">[3]</span></div></li></ol>

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

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

Vector Subtraction

Relationships Between Two Lines

2024 MI PRU1 P1 Q9

<p>With respect to the origin $O$, the position vectors of the points $A$ and $B$ are $\mathbf{i}+2\mathbf{j}+\mathbf{k}$ and $\mathbf{i}-4\mathbf{k}$ respectively. The equation of the planes ${{p}_{1}}$ and ${{p}_{2}}$ are given by ${{p}_{1}}:\mathbf{r}\cdot \left( \begin{matrix}
   1  \\
   3  \\
   0  \\
\end{matrix} \right)=1$ and ${{p}_{2}}:\mathbf{r}\cdot \left( \begin{matrix}
   0  \\
   7  \\
   \alpha   \\
\end{matrix} \right)=-4\alpha $, where&nbsp;$\alpha $ is a real constant with $\alpha &gt;0$.<br></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 position vector of the foot of perpendicular from $A$ to $p_1$.<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">Verify that&nbsp;$B$ lies on both&nbsp;$p_1$ and $p_2$. Show that the vector equation of the line of intersection, $L$, of $p_1$ and $p_2$ is of the form<span class="inline-marks">[2]</span></div></li></ol>
<p style="text-align: center">$\mathbf{r}=\left( \begin{matrix}
   1  \\
   0  \\
   -4  \\
\end{matrix} \right)+s\left( \begin{matrix}
   3\alpha   \\
   -\alpha   \\
   7  \\
\end{matrix} \right)$, $s\in \mathbb{R}$.<br><span class="inline-marks">[2]</span></p>
<ol class="i-list" start="3"><li data-list-item-id="item-iii" data-list-item-path="item-iii"><ol class="abc-list"><li data-list-item-id="item-iii-a" data-list-item-path="item-iii/item-iii-a"><div class="list-item-paragraph">The point $C(-1, 1, -4)$ is equidistant from both&nbsp;$p_1$ and $p_2$. Calculate the value of $\alpha$ for this case. Hence find in scalar product form the equation of the plane $p_3$ in which both the line $L$ and the point $C$ lie.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-iii-b" data-list-item-path="item-iii/item-iii-b"><div class="list-item-paragraph">The plane $p_4$ has equation $y - z = 2$. Given that the angle between $L$ and $p_4$ is $\frac{\pi}{6}$, find the value of $\alpha$.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

With respect to the origin $O$, the position vectors of the points $A$ and $B$ are $\mathbf{i}+2\mathbf{j}+\mathbf{k}$ and $\mathbf{i}-4\mathbf{k}$ respectively. The equation of the planes ${{p}_{1}}$ and ${{p}_{2}}$ are given by ${{p}_{1}}:\mathbf{r}\cdot \left( \begin{matrix}
   1  \\
   3  \\
   0  \\
\end{matrix} \right)=1$ and ${{p}_{2}}:\mathbf{r}\cdot \left( \begin{matrix}
   0  \\
   7  \\
   \alpha   \\
\end{matrix} \right)=-4\alpha $, where $\alpha $ is a real constant with $\alpha >0$.


Find the position vector of the foot of perpendicular from $A$ to $p_1$.

Verify that $B$ lies on both $p_1$ and $p_2$. Show that the vector equation of the line of intersection, $L$, of $p_1$ and $p_2$ is of the form

$\mathbf{r}=\left( \begin{matrix}
   1  \\
   0  \\
   -4  \\
\end{matrix} \right)+s\left( \begin{matrix}
   3\alpha   \\
   -\alpha   \\
   7  \\
\end{matrix} \right)$, $s\in \mathbb{R}$.


2023 ASRJC Promo Q11

<p>Find the coordinates of the foot of perpendicular of point $3\mathbf{i}+\mathbf{j}-9\mathbf{k}$ to the plane $3x+y-z=-3$</p>

Find the coordinates of the foot of perpendicular of point $3\mathbf{i}+\mathbf{j}-9\mathbf{k}$ to the plane $3x+y-z=-3$

<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 $p$.

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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMDUwNjAwNjctbnh2bGczeTFkeDgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDg4OCwiZXhwIjoxNzkzNTA1MTg4fQ.Hyjwf8W-qRJd4R02W1u52UCARUXAgEyibwiPUKr2hvI" 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 lines $l_1$ and $l_2$ have equations</p>
<p style="text-align: center">\[l_1 : \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\0 \end{pmatrix} + \lambda \begin{pmatrix} -2 \\ 1 \\ 2 \end{pmatrix}, \\\quad l_2 : \mathbf{r} = \begin{pmatrix} 0 \\ 5 \\ -8 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ -2 \\ 3 \end{pmatrix}\]</p>
<p>where&nbsp;$\lambda$ and $\mu$ are parameters.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$l_1$ and $l_2$ are skew lines.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>The point $P$ has coordinates $(1, 5, 0)$ and the plane $\pi_1$ has equation $y - z - 2 = 0$.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the coordinates of $P'$, the reflection of $P$ in $\pi_1$.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the cartesian equation of the plane $\pi_2$ containing $P$ and $l_1$.</span><span class="inline-marks">[3]</span></div></li></ol>

The lines $l_1$ and $l_2$ have equations

\[l_1 : \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\0 \end{pmatrix} + \lambda \begin{pmatrix} -2 \\ 1 \\ 2 \end{pmatrix}, \\\quad l_2 : \mathbf{r} = \begin{pmatrix} 0 \\ 5 \\ -8 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ -2 \\ 3 \end{pmatrix}\]

where $\lambda$ and $\mu$ are parameters.

Show that $l_1$ and $l_2$ are skew lines.

The point $P$ has coordinates $(1, 5, 0)$ and the plane $\pi_1$ has equation $y - z - 2 = 0$.

Find the coordinates of $P'$, the reflection of $P$ in $\pi_1$.

Find the cartesian equation of the plane $\pi_2$ containing $P$ and $l_1$.

Math Question Bank