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N(A) E Math

PSLE

PSLE Foundation Mathematics

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

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

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

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

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

Relationships Between a Point and a Plane

Distance from a Point to a Plane

Equations of Planes

Angle Between Two Planes

Line of Intersection of Two Planes

Distance from a Point to a Line

2026 RI P1 Q11

Vectors

<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

<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

<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 $p_1$.

Perpendicular Projection and Reflection in a Plane

Equations of Lines

2025 MI PU1 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 $q'$.

Relationships Between a Line and a Plane

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 $x - 4y + 5z = 12$.

Angle Between a Line and a Plane

2024 NYJC Promo Q9

<p>$\mathbf{v}$ and $\mathbf{n}$ are unit vectors in $3$-dimensional space, with $\mathbf{v} \cdot \mathbf{n} \neq 0$.</p>
<p>Let&nbsp;$P$ be a plane with unit normal $\mathbf{n}$. The reflection of $\mathbf{v}$ in plane $P$, $\mathbf{w}$, is given by the formula\[\mathbf{w} = \mathbf{v} - 2(\mathbf{v} \cdot \mathbf{n})\mathbf{n} . \quad (*)\]</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Calculate the vector $\mathbf{w}$ when&nbsp;$\mathbf{v} = \begin{pmatrix} 0 \\ 0 \\ -1 \end{pmatrix}$ and $\mathbf{n} = \frac{1}{5} \begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix}$.<span class="inline-marks">[2]</span></div></li></ol>
<p>When the unit vectors $\mathbf{v}$ and $\mathbf{n}$ vary, the formula $(*)$ will always result in a unit vector for&nbsp;$\mathbf{w}$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Verify that the vector $\mathbf{w}$ calculated in part <strong>(a)</strong> is a unit vector.<span class="inline-marks">[1]</span></div></li></ol>
<p>Before photovoltaic technology allowed for cost-effective direct conversion of sunlight to electrical energy in solar panels, solar power plants relied on arrays of mirrors to concentrate a large area of sunlight into a receiver. In one such plant (see diagram), the receiver is modelled as a single point $C$ with coordinates $(0, 0, 10)$, with mirrors surrounding it. One particular mirror has centre $M$ with coordinates $(-10, -5, 0)$, and its surface is modelled as part of a plane $P$ passing through $M$ with unit normal $\mathbf{n}$.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1779451489366-7psqn7e00y7.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Nzk0NTE0ODkzNjYtN3BzcW43ZTAweTcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDcyMSwiZXhwIjoxNzkzNTA1MDIxfQ.Fj9dSb9pudmtAEHPc8K6wJtO916EcRKkUczRRy76NVo" 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 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzE5MDMyMjE4NzItbXE2dTY1b2VmNS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0NzIxLCJleHAiOjE3OTM1MDUwMjF9.2mTF3_NCCvVNpFuqbRgZMVze6PO7h29bS_NlRL7RFwE" 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

Three-Dimensional Geometry

Relationships Between Two Lines

2024 MI PRU1 P1 Q9

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" data-width="215" data-height="136" data-proportional style="width:215px;height:136px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:215px;--graph-aspect:215 / 136;"></div>
<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 $Q$.

Intersection of a Line and a Plane

2013 DHS P1 Q12

<p>The planes $p$ and $q$ have equations $\mathbf{r}\cdot \left( \begin{matrix}
   1  \\
   1  \\
   1  \\
\end{matrix} \right)=1$ and $\mathbf{r}\cdot \left( \begin{matrix}
   1  \\
   0  \\
   1  \\
\end{matrix} \right)=2$respectively, and meet in the line $l$.<br></p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the acute angle between $p$ and $q$.	<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find a vector equation for&nbsp;$l$.<span class="inline-marks">[2]</span></div></li></ol>

The planes $p$ and $q$ have equations $\mathbf{r}\cdot \left( \begin{matrix}
   1  \\
   1  \\
   1  \\
\end{matrix} \right)=1$ and $\mathbf{r}\cdot \left( \begin{matrix}
   1  \\
   0  \\
   1  \\
\end{matrix} \right)=2$respectively, and meet in the line $l$.


2015 MJC P2 Q4

<p>${{\Pi }_{1}}:\mathbf{r}\cdot \left( \begin{matrix}
   -2  \\
   -3  \\
   1  \\
\end{matrix} \right)=7$, ${{\Pi }_{2}}:2x+3y-z=4$<br></p>

${{\Pi }_{1}}:\mathbf{r}\cdot \left( \begin{matrix}
   -2  \\
   -3  \\
   1  \\
\end{matrix} \right)=7$, ${{\Pi }_{2}}:2x+3y-z=4$


Relationships Between Two Planes

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMDUwNjAwNjctbnh2bGczeTFkeDgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDcyMSwiZXhwIjoxNzkzNTA1MDIxfQ.mGArXsE7coMIwXm61YtXYLX8Hu5pGe2NSSuxnebRIJ0" 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>Two planes $p_1$ and $p_2$ have respective cartesian equations given by</p>
<p style="text-align: center">\[4x + y + z = 8 \text{ and}\hspace{0.5em} 4x + 3y - z = 0.\]</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the sine of the acute angle between $p_1$ and $p_2$ in the form $\frac{m}{\sqrt{n}}$ where&nbsp;$m$ and $n$ are positive integers to be determined.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Verify that the point $A$ with coordinates $(1,0,4)$ lies on $p_1$ and $p_2$. Hence, without the use of a calculator, find the vector equation of the line $l$ formed by the intersection of $p_1$ and $p_2$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">It is given that&nbsp;$B$ is a point on $p_1$ with coordinates $(1,3,1)$. Show that&nbsp;$AB$ is perpendicular to $l$ and hence use <strong>(a)</strong> to deduce exactly the shortest distance from $B$ to $p_2$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the cartesian equation of the plane $p_3$ which contains $B$ and is perpendicular to both&nbsp;$p_1$ and $p_2$.</span><span class="inline-marks">[2]</span></div></li></ol>

Two planes $p_1$ and $p_2$ have respective cartesian equations given by

\[4x + y + z = 8 \text{ and } 4x + 3y - z = 0.\]

Find the sine of the acute angle between $p_1$ and $p_2$ in the form $\frac{m}{\sqrt{n}}$ where $m$ and $n$ are positive integers to be determined.

Verify that the point $A$ with coordinates $(1,0,4)$ lies on $p_1$ and $p_2$. Hence, without the use of a calculator, find the vector equation of the line $l$ formed by the intersection of $p_1$ and $p_2$.

Find the cartesian equation of the plane $p_3$ which contains $B$ and is perpendicular to both $p_1$ and $p_2$.

Perpendicular Vectors Using the Scalar Product

2025 NYJC P2 Q5

<p>A designer wants to construct an inclined roof that is attached to a vertical rectangular wall, positioned above a horizontal flat ground, as shown in the diagram below. Assume that the roof and wall are of negligible thickness.</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/1768719798428-vepnc083dwo.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg3MTk3OTg0MjgtdmVwbmMwODNkd28ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDcyMSwiZXhwIjoxNzkzNTA1MDIxfQ.r1-OpNkW_cH26Ur0_jRn85REeFqNBNJm0XraM8SyBlw" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Points $(x, y, z)$ are defined relative to a corner of the wall at $O\,\mathrm (0, 0, 0)$, where units are metres. Unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are in the directions shown in the diagram. The other corners of the wall are $A$, $B$ and $C$, such that&nbsp;$AB$ is $5$ m and $A$ and $B$ are $3$ m vertically above $O$ and $C$ respectively. The roof is attached to the wall along $AB$ and is modelled by a plane parallel to $4\mathbf{i} + 5\mathbf{j} + \mathbf{k}$.</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 the roof is defined by the plane $x - 4z = -12$.<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 obtuse angle between the wall and the roof, correct to the nearest degree.<span class="inline-marks">[3]</span></div></li></ol>
<p>To support the roof, a wooden strut, of negligible thickness, is constructed. It has one end at point $P$ with coordinates $(4, 3, 0)$ on the horizontal ground and the other end at the roof, such that the length of the wooden strut is a minimum.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the exact length of the wooden strut.<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">A ray of light is emitted in the direction $7\mathbf{i} - 4\mathbf{j} + 4\mathbf{k}$ from a light source at $C$. Show that this light ray meets the wooden strut. Hence, find the coordinates of the point where the light ray meets the wooden strut.<span class="inline-marks">[4]</span></div></li></ol>

A designer wants to construct an inclined roof that is attached to a vertical rectangular wall, positioned above a horizontal flat ground, as shown in the diagram below. Assume that the roof and wall are of negligible thickness.

Points $(x, y, z)$ are defined relative to a corner of the wall at $O\,\mathrm (0, 0, 0)$, where units are metres. Unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are in the directions shown in the diagram. The other corners of the wall are $A$, $B$ and $C$, such that $AB$ is $5$ m and $A$ and $B$ are $3$ m vertically above $O$ and $C$ respectively. The roof is attached to the wall along $AB$ and is modelled by a plane parallel to $4\mathbf{i} + 5\mathbf{j} + \mathbf{k}$.

Show that the roof is defined by the plane $x - 4z = -12$.

Find the obtuse angle between the wall and the roof, correct to the nearest degree.

To support the roof, a wooden strut, of negligible thickness, is constructed. It has one end at point $P$ with coordinates $(4, 3, 0)$ on the horizontal ground and the other end at the roof, such that the length of the wooden strut is a minimum.

Find the exact length of the wooden strut.

A ray of light is emitted in the direction $7\mathbf{i} - 4\mathbf{j} + 4\mathbf{k}$ from a light source at $C$. Show that this light ray meets the wooden strut. Hence, find the coordinates of the point where the light ray meets the wooden strut.

Intersection of Two Lines

2025 JPJC P1 Q12

<p>The planes ${{\mathrm{p}}_{1}}$ and ${{\mathrm{p}}_{2}}$ have equations $\mathbf{r}\cdot \left( \begin{matrix}1  \\ -3  \\  2  \\\end{matrix} \right)=-5$ and $\mathbf{r}\cdot \left( \begin{matrix}  2  \\-1  \\  3  \\\end{matrix} \right)=1$ respectively, and meet in the line ${{l}_{4}}$. Find</p>
<ol class="i-list"><li><div class="list-item-paragraph">the vector equation for ${{l}_{4}}$,</div></li><li><div class="list-item-paragraph">Find the acute angle between ${{\mathrm{p}}_{1}}$ and ${{\mathrm{p}}_{2}}$.</div></li></ol>

The planes ${{\mathrm{p}}_{1}}$ and ${{\mathrm{p}}_{2}}$ have equations $\mathbf{r}\cdot \left( \begin{matrix}1  \\ -3  \\  2  \\\end{matrix} \right)=-5$ and $\mathbf{r}\cdot \left( \begin{matrix}  2  \\-1  \\  3  \\\end{matrix} \right)=1$ respectively, and meet in the line ${{l}_{4}}$. Find

Unit 13.2 Worked Example 5

<p>The diagram shows a pyramid with horizontal rectangular base $ABCD$, where&nbsp;$AB=8$ cm, $BC=6$ cm. The vertex $V$ of the pyramid is vertically above the centre of the base and the length of $VA$ is $9$ cm. The points $E$, $F$ are the mid-points of $VA$, $VD$ respectively. The point $N$ is on $VB$ such that&nbsp;$VN:NB=2:1$.<br>Giving each answer correct to $3$ significant, figures, find</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--1-2" data-layout="1-2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li data-list-item-id="item-1780162733442-bmaxc36h6" data-list-item-path="item-1780162733442-bmaxc36h6"><div class="list-item-paragraph">the angle between plane $EFCB$ and the base</div></li><li data-list-item-id="item-1780162734386-ld2pqmao0" data-list-item-path="item-1780162734386-ld2pqmao0"><div class="list-item-paragraph">the angle which $CN$ makes with the base.</div></li></ol></div><div class="question-column" style="flex: 2; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="174" data-height="162" data-proportional style="width:174px;height:162px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:174px;--graph-aspect:174 / 162;"></div></div></div></div>

The diagram shows a pyramid with horizontal rectangular base $ABCD$, where $AB=8$ cm, $BC=6$ cm. The vertex $V$ of the pyramid is vertically above the centre of the base and the length of $VA$ is $9$ cm. The points $E$, $F$ are the mid-points of $VA$, $VD$ respectively. The point $N$ is on $VB$ such that $VN:NB=2:1$.
Giving each answer correct to $3$ significant, figures, find

<p>A pyramid with a square base $ABCD$ is as shown in the diagram below. The dimensions are $AB=BC=CD=DA=4$ cm and $VA=VB=VC=VD=7$ cm. $N$ is a point at the foot of the perpendicular from $A$ to $VB$.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li data-list-item-id="item-1780162614868-yu01ylsps" data-list-item-path="item-1780162614868-yu01ylsps"><div class="list-item-paragraph">Show that the length of $VN$ is $5.86$ cm.</div></li><li data-list-item-id="item-1780162615818-isktwdaxd" data-list-item-path="item-1780162615818-isktwdaxd"><div class="list-item-paragraph">Find the angle between the plane $ANC$ and the horizontal.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="150" data-height="154" data-proportional style="width:150px;height:154px;max-width:100%;display:flow-root;margin:0 0 0 auto;--graph-width:150px;--graph-aspect:150 / 154;"></div></div></div></div>

A pyramid with a square base $ABCD$ is as shown in the diagram below. The dimensions are $AB=BC=CD=DA=4$ cm and $VA=VB=VC=VD=7$ cm. $N$ is a point at the foot of the perpendicular from $A$ to $VB$.

Perpendicular Projection and Reflection in a Line

<p>${{\Pi }_{1}}:\mathbf{r}=\left( \begin{matrix}1  \\3  \\1  \\\end{matrix} \right)+\lambda \left( \begin{matrix} 2  \\-2  \\ 1  \\\end{matrix} \right)+\mu \left( \begin{matrix}1  \\ 5  \\3  \\\end{matrix} \right);\lambda ,\mu \in \mathbb{R}$, </p>
<p>${{\Pi }_{2}}:(2t+3s)\mathbf{i}+(1-2t+3s)\mathbf{j}+(-2+t+4s)\mathbf{k};s,t\in \mathbb{R}$</p>

${{\Pi }_{1}}:\mathbf{r}=\left( \begin{matrix}1  \\3  \\1  \\\end{matrix} \right)+\lambda \left( \begin{matrix} 2  \\-2  \\ 1  \\\end{matrix} \right)+\mu \left( \begin{matrix}1  \\ 5  \\3  \\\end{matrix} \right);\lambda ,\mu \in \mathbb{R}$, 

${{\Pi }_{2}}:(2t+3s)\mathbf{i}+(1-2t+3s)\mathbf{j}+(-2+t+4s)\mathbf{k};s,t\in \mathbb{R}$

<p>Find the relationship and the acute angles between the two planes. If the planes are parallel to each other, find the perpendicular distance between the planes, otherwise, find the equation of line which two planes intersect.</p>
<p>${{\Pi }_{1}}:\mathbf{r}=\left( \begin{matrix} 1\\ 0\\ -3 \\\end{matrix} \right)+\lambda \left( \begin{matrix} -2 \\1\\ 0 \\\end{matrix} \right)+\mu \left( \begin{matrix} 1\\ -2\\ 3 \\\end{matrix} \right);\lambda ,\mu \in \mathbb{R}$, </p>
<p>${{\Pi }_{2}}:(-1+2t-3s)\mathbf{i}+(-3-t+s)\mathbf{j}+(5+2t-s)\mathbf{k};s,t\in \mathbb{R}$</p>

Find the relationship and the acute angles between the two planes. If the planes are parallel to each other, find the perpendicular distance between the planes, otherwise, find the equation of line which two planes intersect.

${{\Pi }_{1}}:\mathbf{r}=\left( \begin{matrix} 1\\ 0\\ -3 \\\end{matrix} \right)+\lambda \left( \begin{matrix} -2 \\1\\ 0 \\\end{matrix} \right)+\mu \left( \begin{matrix} 1\\ -2\\ 3 \\\end{matrix} \right);\lambda ,\mu \in \mathbb{R}$, 

${{\Pi }_{2}}:(-1+2t-3s)\mathbf{i}+(-3-t+s)\mathbf{j}+(5+2t-s)\mathbf{k};s,t\in \mathbb{R}$

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