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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>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 Line and a Plane

Intersection of a Line and a Plane

2026 HCI P1 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

<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 $p_1$.

Equations of Lines

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 $\pi_1$ and $\pi_2$, which meet in the line $l_1$, have equations</p>
<p style="text-align: center">\[\pi_1 : \mathbf{r} \cdot \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix} = 25 \quad \text{and} \quad \pi_2 : x + ky - 2z = -15,\]</p>
<p>where&nbsp;$k$ is a constant.</p>
<p>Another line $l_2$ has equation $\mathbf{r} = \begin{pmatrix} -15 \\ 0 \\ 0 \end{pmatrix} + \beta \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix}, \beta \in \mathbb{R}$.</p>
<ol class="i-list"><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">Determine the position vector of the point on $l_1$ such that the coordinates of this point is independent of $k$. Hence find a vector equation of $l_1$.<span class="inline-marks">[4]</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">Determine the possible value(s) of $k$ such that&nbsp;$l_1$ and $l_2$ are skew.<span class="inline-marks">[3]</span></div></li></ol>
<p>Assume that&nbsp;$k = 4$ for the rest of this question.</p>
<ol class="i-list" start="3"><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">Points $A$ and $B$ are on $l_1$ and $l_2$ respectively such that&nbsp;$\overrightarrow{AB}$ is perpendicular to both lines. Show that&nbsp;$\left\vert{} \overrightarrow{AB} \right\vert{} = \sqrt{\frac{p}{2}}$, where&nbsp;$p$ is an integer to be determined.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-6-item-1" data-list-item-path="imported-item-block-6-item-1"><div class="list-item-paragraph">Find exactly the sine of the acute angle between $l_2$ and $\pi_1$.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is given further that&nbsp;$l_2$ lies on a third plane that is perpendicular to $l_1$, and $l_2$ intersects $\pi_1$ at point $P$.</p>
<ol class="i-list" start="5"><li data-list-item-id="imported-item-block-8-item-0" data-list-item-path="imported-item-block-8-item-0"><div class="list-item-paragraph">Deduce the shortest distance from $P$ to $l_1$.<span class="inline-marks">[2]</span></div></li></ol>

The planes $\pi_1$ and $\pi_2$, which meet in the line $l_1$, have equations

\[\pi_1 : \mathbf{r} \cdot \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix} = 25 \quad \text{and} \quad \pi_2 : x + ky - 2z = -15,\]

Another line $l_2$ has equation $\mathbf{r} = \begin{pmatrix} -15 \\ 0 \\ 0 \end{pmatrix} + \beta \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix}, \beta \in \mathbb{R}$.

Determine the position vector of the point on $l_1$ such that the coordinates of this point is independent of $k$. Hence find a vector equation of $l_1$.

Determine the possible value(s) of $k$ such that $l_1$ and $l_2$ are skew.

Assume that $k = 4$ for the rest of this question.

Points $A$ and $B$ are on $l_1$ and $l_2$ respectively such that $\overrightarrow{AB}$ is perpendicular to both lines. Show that $\left\vert{} \overrightarrow{AB} \right\vert{} = \sqrt{\frac{p}{2}}$, where $p$ is an integer to be determined.

Find exactly the sine of the acute angle between $l_2$ and $\pi_1$.

It is given further that $l_2$ lies on a third plane that is perpendicular to $l_1$, and $l_2$ intersects $\pi_1$ at point $P$.

Deduce the shortest distance from $P$ to $l_1$.

Finding Points on a Plane

Common Perpendicular and Distance Between Skew Lines

Relationships Between Two Lines

Perpendicular Vectors Using the Scalar Product

2024 NJC P1 Q12

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

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

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

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

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

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

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

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

Perpendicular Projection and Reflection in a Line

<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>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 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>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>The planes $p_1$ and $p_2$ have equations $\mathbf{r} \cdot \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix} = 0$ and $\mathbf{r} \cdot \begin{pmatrix} 2 \\ -1 \\ -2 \end{pmatrix} = 6$ respectively.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find a vector equation of $l$, the line of intersection between $p_1$ and $p_2$.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>Another line, $l_1$, has equation $\mathbf{r} = \begin{pmatrix} 2 \\ 0 \\ 8 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 1 \\ 5 \end{pmatrix}$, where&nbsp;$\mu$ is a real parameter.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Use the result in part (i) to show that&nbsp;$p_1$, $p_2$ and $l_1$ have exactly one common point of intersection. (You are not allowed to use any calculator for this part of the question.)</span><span class="inline-marks">[3]</span></div></li></ol>

The planes $p_1$ and $p_2$ have equations $\mathbf{r} \cdot \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix} = 0$ and $\mathbf{r} \cdot \begin{pmatrix} 2 \\ -1 \\ -2 \end{pmatrix} = 6$ respectively.

Find a vector equation of $l$, the line of intersection between $p_1$ and $p_2$.

Another line, $l_1$, has equation $\mathbf{r} = \begin{pmatrix} 2 \\ 0 \\ 8 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 1 \\ 5 \end{pmatrix}$, where $\mu$ is a real parameter.

Use the result in part (i) to show that $p_1$, $p_2$ and $l_1$ have exactly one common point of intersection. (You are not allowed to use any calculator for this part of the question.)

Intersection of Two Lines

2015 NJC P2 Q3

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

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

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

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

Areas of Triangles and Parallelograms Using Vectors

2025 YIJC P1 Q10

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1769305060067-nxvlg3y1dx8.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMDUwNjAwNjctbnh2bGczeTFkeDgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgyNCwiZXhwIjoxNzkzNTA1MTI0fQ.z3__HsgXeU5n5399l4tTVLKUosEqofC46Brbi80X5Wg" 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 $p_2$.

2025 NYJC P2 Q5

<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><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$c=-1$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">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><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">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><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the equation of the plane $\pi_3$ which is parallel to $\pi_2$ and contains the point $B$. Hence show that the distance between the planes $\pi_2$ and $\pi_3$ is $\frac{20}{3\sqrt{3}}$.</span><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.

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 $\alpha$ is a parameter.

Find the position vectors of the points on the line $l$ which are a distance of $3\sqrt{2}$ from the point $B(2, -3, 7)$.

Find the equation of the plane $\pi_3$ which is parallel to $\pi_2$ and contains the point $B$. Hence show that the distance between the planes $\pi_2$ and $\pi_3$ is $\frac{20}{3\sqrt{3}}$.

<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>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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