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IB AA HL Math

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N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>Underwater sonar buoys are deployed, in the ocean, to detect and monitor threats in the ocean and on the ocean surface. Two underwater sonar buoys, buoy A and buoy B, are located at points $A$ and $B$ respectively. The underwater sonar buoys are in the water body.</p>
<p>Points are defined relative to an origin $(0,0,0)$ on the ocean bed, with units in metres.</p>
<p>The ocean surface is modelled as the plane with equation $3y+4z=240$. A submarine moves along a straight path with equation $\frac x2=y-20=\frac{-z+13}2$. $A$ and $B$ have coordinates $(-10,31,5)$ and $(4,17,19)$ respectively.</p>
<ol class="abc-list"><li data-list-item-id="rvhs-q12-l4-item01" data-list-item-path="rvhs-q12-l4-item01"><div class="list-item-paragraph">Find a cartesian equation of the plane containing the path of the submarine and buoy B.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="rvhs-q12-l4-item02" data-list-item-path="rvhs-q12-l4-item02"><div class="list-item-paragraph">Determine the acute angle between the path of the submarine and the ocean surface.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="rvhs-q12-l4-item03" data-list-item-path="rvhs-q12-l4-item03"><div class="list-item-paragraph">An unidentified boat on the ocean surface is detected by buoys A and B. The boat is found to be equidistant from both buoys A and B. Describe geometrically the set of all possible positions of the location of the unidentified boat.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="rvhs-q12-l4-item04" data-list-item-path="rvhs-q12-l4-item04"><div class="list-item-paragraph">At low tide, the ocean surface falls below the current ocean surface at a perpendicular distance of $0.8$ metres. The ocean surface at low tide is also modelled as a plane. Find the cartesian equation of this plane.<span class="inline-marks">[2]</span></div></li></ol>

Underwater sonar buoys are deployed, in the ocean, to detect and monitor threats in the ocean and on the ocean surface. Two underwater sonar buoys, buoy A and buoy B, are located at points $A$ and $B$ respectively. The underwater sonar buoys are in the water body.

Points are defined relative to an origin $(0,0,0)$ on the ocean bed, with units in metres.

The ocean surface is modelled as the plane with equation $3y+4z=240$. A submarine moves along a straight path with equation $\frac x2=y-20=\frac{-z+13}2$. $A$ and $B$ have coordinates $(-10,31,5)$ and $(4,17,19)$ respectively.

Find a cartesian equation of the plane containing the path of the submarine and buoy B.

Determine the acute angle between the path of the submarine and the ocean surface.

An unidentified boat on the ocean surface is detected by buoys A and B. The boat is found to be equidistant from both buoys A and B. Describe geometrically the set of all possible positions of the location of the unidentified boat.

At low tide, the ocean surface falls below the current ocean surface at a perpendicular distance of $0.8$ metres. The ocean surface at low tide is also modelled as a plane. Find the cartesian equation of this plane.

Distance from a Point to a Plane

Angle Between a Line and a Plane

Equations of Planes

Geometrical Interpretation of Vector Equations

Line of Intersection of Two Planes

2026 RVHS P1 Q12

Vectors

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

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

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

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

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

Relationships Between a Point and a Plane

Angle Between Two Planes

Distance from a Point to a Line

2026 RI P1 Q11

<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 $\pi_2$.

Intersection of a Line and a Plane

2026 TJC P1 Q8

<p>The lines $l_1$ and $l_2$ have equations \[\mathbf{r}=\begin{pmatrix}2\\1\\-1\end{pmatrix}+\lambda\begin{pmatrix}1\\-2\\3\end{pmatrix}\qquad\text{and}\qquad\mathbf{r}=\begin{pmatrix}3\\4\\p\end{pmatrix}+\mu\begin{pmatrix}2\\1\\q\end{pmatrix}\] respectively, where&nbsp;$p$ and $q$ are constants, and $\lambda$ and $\mu$ are parameters.</p>
<ol class="abc-list"><li data-list-item-id="nyjc-q11-item01" data-list-item-path="nyjc-q11-item01"><div class="list-item-paragraph">Suppose that&nbsp;$l_1$ intersects $l_2$.</div><ol class="i-list"><li data-list-item-id="nyjc-q11-item01-sub-item01" data-list-item-path="nyjc-q11-item01/nyjc-q11-item01-sub-item01"><div class="list-item-paragraph">Find the coordinates of the point of intersection and state a linear relationship between $p$ and $q$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="nyjc-q11-item01-sub-item02" data-list-item-path="nyjc-q11-item01/nyjc-q11-item01-sub-item02"><div class="list-item-paragraph">Find the acute angle between $l_1$ and $l_2$ when&nbsp;$q=2$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="nyjc-q11-item02" data-list-item-path="nyjc-q11-item02"><div class="list-item-paragraph">By using vector product, show that a vector perpendicular to both&nbsp;$l_1$ and $l_2$ is $\begin{pmatrix}2q+3\\q-6\\m\end{pmatrix}$, where&nbsp;$m$ is a constant to be determined.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="nyjc-q11-item03" data-list-item-path="nyjc-q11-item03"><div class="list-item-paragraph">The plane $\pi$ contains $l_2$ and is parallel to $l_1$. Find the shortest distance between $l_1$ and $\pi$ in terms of $p$ and $q$.<span class="inline-marks">[4]</span></div></li></ol>

The lines $l_1$ and $l_2$ have equations \[\mathbf{r}=\begin{pmatrix}2\\1\\-1\end{pmatrix}+\lambda\begin{pmatrix}1\\-2\\3\end{pmatrix}\qquad\text{and}\qquad\mathbf{r}=\begin{pmatrix}3\\4\\p\end{pmatrix}+\mu\begin{pmatrix}2\\1\\q\end{pmatrix}\] respectively, where $p$ and $q$ are constants, and $\lambda$ and $\mu$ are parameters.

By using vector product, show that a vector perpendicular to both $l_1$ and $l_2$ is $\begin{pmatrix}2q+3\\q-6\\m\end{pmatrix}$, where $m$ is a constant to be determined.

The plane $\pi$ contains $l_2$ and is parallel to $l_1$. Find the shortest distance between $l_1$ and $\pi$ in terms of $p$ and $q$.

Angles Between Two Lines

Intersection of Two Lines

Vector (Cross) Product

2026 NYJC P1 Q11

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<p>With reference to the origin $O$, the points $A$, $B$ and $C$ are such that vectors $\overrightarrow{OA}=\mathbf{a}$, $\overrightarrow{OB}=\mathbf{b}$ and $\overrightarrow{OC}=2\mathbf{a}-2\mathbf{b}$.</p>
<ol class="abc-list"><li data-list-item-id="acjc-q08-item01" data-list-item-path="acjc-q08-item01"><div class="list-item-paragraph">Given that&nbsp;$|\mathbf{a}|=2$, $|\mathbf{b}|=3$ and the angle between $\mathbf{a}$ and $\mathbf{b}$ is $\frac{\pi}{6}$, find the area of triangle $OBC$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="acjc-q08-item02" data-list-item-path="acjc-q08-item02"><div class="list-item-paragraph">The lines $OA$ and $BC$ intersect at point $X$.</div><div class="list-item-paragraph">Show that&nbsp;$\displaystyle \overrightarrow{OX}=\frac23\mathbf{a}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="acjc-q08-item03" data-list-item-path="acjc-q08-item03"><div class="list-item-paragraph">The plane $p$ has equation $\mathbf{r}\cdot\mathbf{n}=d$ and contains the point $A$, where&nbsp;$\mathbf{n}$ is a unit vector.</div><div class="list-item-paragraph">Find the shortest distance from point $X$ to plane $p$, in the form of $\frac1k|d|$, where&nbsp;$k$ is a positive integer to be determined.<span class="inline-marks">[3]</span></div></li></ol>

With reference to the origin $O$, the points $A$, $B$ and $C$ are such that vectors $\overrightarrow{OA}=\mathbf{a}$, $\overrightarrow{OB}=\mathbf{b}$ and $\overrightarrow{OC}=2\mathbf{a}-2\mathbf{b}$.

Given that $|\mathbf{a}|=2$, $|\mathbf{b}|=3$ and the angle between $\mathbf{a}$ and $\mathbf{b}$ is $\frac{\pi}{6}$, find the area of triangle $OBC$.

Areas of Triangles and Parallelograms Using Vectors

2026 ACJC P1 Q8

<div class="graph-block" 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data-width="420" data-height="360" data-proportional style="width:420px;height:360px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:420px;--graph-aspect:420 / 360;"></div>
<p>An oblique pyramid has a plane base $ABCD$ in the shape of a parallelogram. The coordinates of $A$, $B$ and $C$ are $(5,-4,1)$, $(5,4,0)$ and $(-5,4,2)$ respectively. The apex of the pyramid is at $E(0,0,10)$ (see diagram).</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">Find the coordinates of $D$.<span class="inline-marks">[1]</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">Find the cartesian equation of face $BCE$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-2-item-2" data-list-item-path="imported-item-block-2-item-2"><div class="list-item-paragraph">Find the angle between face $BCE$ and the base of the pyramid.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-2-item-3" data-list-item-path="imported-item-block-2-item-3"><div class="list-item-paragraph">Find the shortest distance from the midpoint of edge $AD$ to face $BCE$.<span class="inline-marks">[5]</span></div></li></ol>

An oblique pyramid has a plane base $ABCD$ in the shape of a parallelogram. The coordinates of $A$, $B$ and $C$ are $(5,-4,1)$, $(5,4,0)$ and $(-5,4,2)$ respectively. The apex of the pyramid is at $E(0,0,10)$ (see diagram).

Find the cartesian equation of face $BCE$.

Find the angle between face $BCE$ and the base of the pyramid.

Find the shortest distance from the midpoint of edge $AD$ to face $BCE$.

Vectors in a Parallelogram

Ratio Theorem and Section Formula

N2018 P2 Q3

<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 $\mathbf{a}\cdot\mathbf{b}=3|\mathbf{a}|^2-2|\mathbf{b}|^2$.

Perpendicular Projection and Reflection in a Plane

Perpendicular Vectors Using the Scalar Product

2025 YIJC Promo Q8

<p>The planes $\pi_1$ and $\pi_2$ have equations</p>
<p style="text-align: center">${2x + 2y + z = 2,}$<br>${5x - 2y - z = 5,}$</p>
<p>respectively.</p>
<ol class="i-list"><li data-list-item-id="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-0" data-list-item-path="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-0"><div class="list-item-paragraph">Find the acute angle between $\pi_1$ and $\pi_2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-1" data-list-item-path="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-1"><div class="list-item-paragraph">Verify that the point $P$ with position vector $\mathbf{i}+\mathbf{j}-2\mathbf{k}$ lies in $\pi_1$ and $\pi_2$. Hence, find a vector equation of the line of intersection of $\pi_1$ and $\pi_2$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-2" data-list-item-path="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-2"><div class="list-item-paragraph">The point $Q$ has position vector $\alpha\mathbf{i}+2\alpha\mathbf{k}$, where&nbsp;$\alpha \in \mathbb{R}$. Show that the perpendicular distance from $Q$ to $\pi_1$ is $\dfrac{2|2\alpha-1|}{3}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-3" data-list-item-path="h2-2025-sajc-promo-q10-sajc-q10-item-block-3-item-3"><div class="list-item-paragraph"><p>Hence or otherwise, find the cartesian equations of the two planes that are parallel to $\pi_1$ and at a distance of $6$ units from $\pi_1$.</p><span class="inline-marks">[4]</span></div></li></ol>

The planes $\pi_1$ and $\pi_2$ have equations

${2x + 2y + z = 2,}$
${5x - 2y - z = 5,}$

Find the acute angle between $\pi_1$ and $\pi_2$.

Verify that the point $P$ with position vector $\mathbf{i}+\mathbf{j}-2\mathbf{k}$ lies in $\pi_1$ and $\pi_2$. Hence, find a vector equation of the line of intersection of $\pi_1$ and $\pi_2$.

The point $Q$ has position vector $\alpha\mathbf{i}+2\alpha\mathbf{k}$, where $\alpha \in \mathbb{R}$. Show that the perpendicular distance from $Q$ to $\pi_1$ is $\dfrac{2|2\alpha-1|}{3}$.

2025 SAJC Promo Q10

<div class="graph-block" data-graph-config="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" data-width="352" data-height="258" style="width:352px;height:258px;max-width:100%;display:block;margin:0 auto;"></div>
<p>The diagram shows a tetrahedron with vertices $O$, $A$, $B$ and $C$. Points $A$, $B$ and $C$ have position vectors with respect to the origin $O$ given by $\overrightarrow{OA}=\begin{pmatrix}2\\-1\\2\end{pmatrix}$, $\overrightarrow{OB}=\begin{pmatrix}0\\3\\1\end{pmatrix}$ and $\overrightarrow{OC}=\begin{pmatrix}3\\0\\4\end{pmatrix}$. The point $D$ lies on $BC$, between $B$ and $C$, such that&nbsp;$BD:DC=1:2$.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-nyjc-promo-q9-nyjc-q9-item-block-2-item-0" data-list-item-path="h2-2025-nyjc-promo-q9-nyjc-q9-item-block-2-item-0"><div class="list-item-paragraph">Find the position vector of $D$<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-nyjc-promo-q9-nyjc-q9-item-block-2-item-1" data-list-item-path="h2-2025-nyjc-promo-q9-nyjc-q9-item-block-2-item-1"><div class="list-item-paragraph">Find a cartesian equation of face $ABC$ and state its exact distance from $O$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="h2-2025-nyjc-promo-q9-nyjc-q9-item-block-2-item-2" data-list-item-path="h2-2025-nyjc-promo-q9-nyjc-q9-item-block-2-item-2"><div class="list-item-paragraph">Hence find the acute angle between $OD$ and face $ABC$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-nyjc-promo-q9-nyjc-q9-item-block-2-item-3" data-list-item-path="h2-2025-nyjc-promo-q9-nyjc-q9-item-block-2-item-3"><div class="list-item-paragraph">Find the coordinates of the point on $OD$ which is closest to $A$.<span class="inline-marks">[3]</span></div></li></ol>

The diagram shows a tetrahedron with vertices $O$, $A$, $B$ and $C$. Points $A$, $B$ and $C$ have position vectors with respect to the origin $O$ given by $\overrightarrow{OA}=\begin{pmatrix}2\\-1\\2\end{pmatrix}$, $\overrightarrow{OB}=\begin{pmatrix}0\\3\\1\end{pmatrix}$ and $\overrightarrow{OC}=\begin{pmatrix}3\\0\\4\end{pmatrix}$. The point $D$ lies on $BC$, between $B$ and $C$, such that $BD:DC=1:2$.

Find a cartesian equation of face $ABC$ and state its exact distance from $O$.

Hence find the acute angle between $OD$ and face $ABC$.

Find the coordinates of the point on $OD$ which is closest to $A$.

Perpendicular Projection and Reflection in a Line

2025 NYJC Promo Q9

<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 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 $p$.

Relationships Between a Line and 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 $C$.

<p>Referred to the origin $O$, the points $A$, $B$ and $C$ have position vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively and they lie on plane $\pi$ .</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;$\mathbf{a}\times\mathbf{b} + \mathbf{b}\times\mathbf{c} + \mathbf{c}\times\mathbf{a}$ is a vector perpendicular to the plane $\pi$ .<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Prove that the equation of plane $\pi$ can be written as</div><div class="list-item-paragraph" style="text-align: center">\[\mathbf{r}\bullet(\mathbf{a}\times\mathbf{b} + \mathbf{b}\times\mathbf{c} + \mathbf{c}\times\mathbf{a}) = \mathbf{a}\bullet(\mathbf{b}\times\mathbf{c}),\]</div><div class="list-item-paragraph">explaining clearly the reason for any result that you use in your proof.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Given that&nbsp;$\mathbf{a} = \mathbf{i}$, $\mathbf{b} = \mathbf{j}$ and $\mathbf{c} = \mathbf{k}$, show that the equation of the plane $\pi$ can be written as&nbsp;$\mathbf{r}\bullet\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = 1$. Hence, find the cartesian equations of the planes which are at a distance of 5 units from plane $\pi$.<span class="inline-marks">[3]</span></div></li></ol>

Referred to the origin $O$, the points $A$, $B$ and $C$ have position vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively and they lie on plane $\pi$ .

Scalar (Dot) Product

2024 ACJC P2 Q2

<p>The planes $\pi_1$ and $\pi_2$ have equations $3x+c(y+z)-2=0$ and $\mathbf{r}=(\mathbf{i}+3\mathbf{j}-2\mathbf{k})+s(2\mathbf{i}-\mathbf{j}+3\mathbf{k})+t(\mathbf{i}-\mathbf{k})$ respectively, where&nbsp;$c$ is a constant, and $s$ and $t$ are parameters. The point $A(1,3,-2)$ lies in both planes.</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;$c=-1$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Show that the vector equation of the line of intersection of $\pi_1$ and $\pi_2$, line $l$, is given by $\mathbf{r}=\mathbf{i}+3\mathbf{j}-2\mathbf{k}+\alpha(\mathbf{i}-\mathbf{j}+4\mathbf{k})$, where&nbsp;$\alpha$ is a parameter.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the position vectors of the points on the line $l$ which are a distance of $3\sqrt{2}$ from the point $B(2,-3,7)$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Find the equation of the plane $\pi_3$ which is parallel to $\pi_2$ and contains the point $B$.</div><div class="list-item-paragraph">Hence show that the distance between the planes $\pi_2$ and $\pi_3$ is $\frac{20}{3\sqrt{3}}$.<span class="inline-marks">[3]</span></div></li></ol>

The planes $\pi_1$ and $\pi_2$ have equations $3x+c(y+z)-2=0$ and $\mathbf{r}=(\mathbf{i}+3\mathbf{j}-2\mathbf{k})+s(2\mathbf{i}-\mathbf{j}+3\mathbf{k})+t(\mathbf{i}-\mathbf{k})$ respectively, where $c$ is a constant, and $s$ and $t$ are parameters. The point $A(1,3,-2)$ lies in both planes.

Magnitude of a Vector

2025 ACJC P1 Q9

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

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

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

Three-Dimensional Geometry

2025 VJC P2 Q3

<ol class="abc-list" start="2"><li data-list-item-id="item-1780997990538-ccqpro9r7"><div class="list-item-paragraph">The position vectors of the variable point $R$ and the fixed points $A$ and $B$ are given by</div><div class="list-item-paragraph" style="text-align: center">$\mathbf{r} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \mathbf{a} = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix} \text{ and}\hspace{0.5em} \mathbf{b} = \begin{pmatrix} -1 \\ 5 \\ 3 \end{pmatrix} \text{ respectively.}$</div><div class="list-item-paragraph">It is given that&nbsp;$\mathbf{r} \cdot (\mathbf{a} \times \mathbf{b}) = 0$.</div><ol class="i-list"><li data-list-item-id="subitem-1780997993251-00a0o1uz1"><div class="list-item-paragraph">Describe geometrically the set of points, denoted by $S$, to which $R$ belongs.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1780997993490-ia1hb1u89"><div class="list-item-paragraph">Find the exact shortest distance between the point $(-8, -2, 9)$ and $S$.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

2021 DHS P2 Q5 Modified

<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 $q'$.

2024 RI Promo Q9

<p>The plane $\Pi_1$ has equation $\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}$, where&nbsp;$\lambda$ and $\mu$ are parameters. The plane $\Pi_2$ has equation $x - 4y + 5z = 12$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the acute angle between $\Pi_1$ and $\Pi_2$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find a vector equation of the line of intersection between $\Pi_1$ and $\Pi_2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">The line $l$ passes through the point $A$ with position vector $m\mathbf{i} + (2m + 1)\mathbf{j} - 3\mathbf{k}$ and is parallel to $3n\mathbf{i} - 3\mathbf{j} + n\mathbf{k}$, where&nbsp;$m$ and $n$ are positive constants. Given that the perpendicular distance from $A$ to the plane $\Pi_1$ is $\frac{15}{\sqrt{6}}$ and that the acute angle between $l$ and $\Pi_1$ is $\sin^{-1} \frac{2}{\sqrt{6}}$, find the values of $m$ and $n$.<span class="inline-marks">[6]</span></div></li></ol>

The plane $\Pi_1$ has equation $\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}$, where $\lambda$ and $\mu$ are parameters. The plane $\Pi_2$ has equation $x - 4y + 5z = 12$.

2024 NYJC Promo Q9

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

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

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

2024 NYJC Promo Q5

<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 $(a, 1, 0)$.

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