A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

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

<ol class="abc-list"><li data-list-item-id="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-0" data-list-item-path="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-0"><div class="list-item-paragraph">The non-zero vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ are such that&nbsp;$3\mathbf{a}\times\mathbf{b}=2\mathbf{b}\times\mathbf{c}$, where&nbsp;$3\mathbf{a}\ne -2\mathbf{c}$. Find a linear relationship between $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1" data-list-item-path="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1"><div class="list-item-paragraph"><p>The variable position vector $\mathbf{v}$ satisfies the equation</p></div><div class="list-item-paragraph"><p style="text-align: center;">$\mathbf{v}\times(-\mathbf{i}+2\mathbf{j}+\mathbf{k})=a\mathbf{i}+\mathbf{j}+\mathbf{k},$</p></div><div class="list-item-paragraph"><p>where $a$ is a non-zero constant.</p></div><ol class="i-list"><li data-list-item-id="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1-sub-0" data-list-item-path="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1/h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1-sub-0"><div class="list-item-paragraph">Show that&nbsp;$a=3$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1-sub-1" data-list-item-path="h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1/h2-2025-ri-promo-q6-ri-q6-item-block-0-item-1-sub-1"><div class="list-item-paragraph">Hence, by finding the set of position vectors $\mathbf{v}$, describe geometrically the set of points represented by the position vectors $\mathbf{v}$.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

The non-zero vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ are such that $3\mathbf{a}\times\mathbf{b}=2\mathbf{b}\times\mathbf{c}$, where $3\mathbf{a}\ne -2\mathbf{c}$. Find a linear relationship between $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$.

Parallel Vectors

Equations of Lines

Vector (Cross) Product

Perpendicular Vectors Using the Scalar Product

2025 RI Promo Q6

<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>With respect to the origin $O$, the fixed points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where&nbsp;$\mathbf{a}$ and $\mathbf{b}$ are non-zero and non-parallel.</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">The variable point $R$ has position vector $\mathbf{r} = \lambda \mathbf{a} + (1 - \lambda)\mathbf{b}$, where&nbsp;$\lambda$ is a real parameter. Describe geometrically the set of all possible positions of $R$.<span class="inline-marks">[1]</span></div></li></ol>
<p>It is given that the angle $AOB$ is $90^\circ$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b"><div class="list-item-paragraph">Explain why $\mathbf{a} \cdot \mathbf{b} = 0$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-c"><div class="list-item-paragraph">Among the set of all possible points $R$, the point $R^*$ is the closest to the origin $O$. Find the position vector of $R^*$. Hence state the ratio $AR^* : BR^*$ in terms of magnitudes of $\mathbf{a}$ and $\mathbf{b}$.<span class="inline-marks">[4]</span></div></li></ol>

With respect to the origin $O$, the fixed points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a}$ and $\mathbf{b}$ are non-zero and non-parallel.

It is given that the angle $AOB$ is $90^\circ$.

Perpendicular Projection and Reflection in a Line

Ratio Theorem and Section Formula

2024 RI Promo Q2

<ol class="abc-list"><li data-list-item-id="item-a"><ol class="i-list"><li data-list-item-id="item-a-i"><div class="list-item-paragraph">Given that&nbsp;$\mathbf{p} \cdot \mathbf{q} = 0$, what can be deduced about the vectors $\mathbf{p}$ and $\mathbf{q}$?<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-a-ii"><div class="list-item-paragraph">With respect to the origin on a $xy$-plane, a fixed point $G$ has position vector $\mathbf{g}$ and a variable point $R$ has position vector $\mathbf{r}$. Describe the locus of $R$ if&nbsp;$\mathbf{r} \cdot (\mathbf{r} - \mathbf{g}) = 0$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-b"><div class="list-item-paragraph">With respect to the origin, plane $p$ has an equation given by $\mathbf{r} = \mathbf{i} + \mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}) + \mu(2\mathbf{i} + 3\mathbf{j} - 2\mathbf{k})$, $\lambda \in \mathbb{R}$, $\mu \in \mathbb{R}$. Points $Q$ and $R$ have position vectors $3\mathbf{i} + 9\mathbf{j} + 5\mathbf{k}$ and $4\mathbf{i} + 11\mathbf{j} + 7\mathbf{k}$ respectively. Find the length of the projection of $\overrightarrow{QR}$ onto $p$.<span class="inline-marks">[4]</span></div></li></ol>

Projection Lengths

2024 NJC Promo Q7

<p></p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">The non-zero vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are such that&nbsp;$\mathbf{a} \times \mathbf{b} = \mathbf{c} \times \mathbf{a}$. Given that&nbsp;$\mathbf{b} \neq -\mathbf{c}$, find a linear relationship between $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">The variable position vector $\mathbf{v}$ satisfies the equation $\mathbf{v} \times (\mathbf{i} - 3\mathbf{k}) = 2\mathbf{j}$.</div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the set of position vectors $\mathbf{v}$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Describe the set of points represented by position vectors $\mathbf{v}$ geometrically.</span><span class="inline-marks">[2]</span></div></li></ol></li></ol>

The non-zero vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are such that $\mathbf{a} \times \mathbf{b} = \mathbf{c} \times \mathbf{a}$. Given that $\mathbf{b} \neq -\mathbf{c}$, find a linear relationship between $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$.

2025 Specimen Paper P1 Q6

<p>The plane $p$ passes through the points $A$, $B$ and $C$ with coordinates $(1, 0, 2)$, $(2, -1, 3)$ and $(-4, -1, 0)$ respectively.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that a cartesian equation of $p$ is $x - y - 2z = -3$.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>The line $l$ has equation $\mathbf{r} = \begin{pmatrix} 2 \\ 0 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -1 \\ -1 \end{pmatrix}, \lambda \in \mathbb{R}$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the acute angle between $l$ and $p$.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>It is given that a variable point $R$ lies on $p$ and is at a distance of $\sqrt{22}$ from the point $Q$ with coordinates $(3, 4, -2)$.</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the foot of perpendicular from $Q$ to $p$.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence describe geometrically the path traced by $R$.</span><span class="inline-marks">[2]</span></div></li></ol>

The plane $p$ passes through the points $A$, $B$ and $C$ with coordinates $(1, 0, 2)$, $(2, -1, 3)$ and $(-4, -1, 0)$ respectively.

Show that a cartesian equation of $p$ is $x - y - 2z = -3$.

The line $l$ has equation $\mathbf{r} = \begin{pmatrix} 2 \\ 0 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -1 \\ -1 \end{pmatrix}, \lambda \in \mathbb{R}$.

Find the acute angle between $l$ and $p$.

It is given that a variable point $R$ lies on $p$ and is at a distance of $\sqrt{22}$ from the point $Q$ with coordinates $(3, 4, -2)$.

Find the foot of perpendicular from $Q$ to $p$.

Hence describe geometrically the path traced by $R$.

Perpendicular Projection and Reflection in a Plane

2025 CJC P1 Q8

<p>Referred to the origin $O$, a variable point $R$ has position vector $\mathbf{r}$ and the fixed points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where&nbsp;$\mathbf{a}$ and $\mathbf{b}$ are non-zero vectors.</p>
<ol class="abc-list"><li data-list-item-id="item-1761382793199-x35engpct" data-list-item-path="item-1761382793199-x35engpct"><div class="list-item-paragraph">If&nbsp;$\mathbf{a}$ and $\mathbf{b}$ are parallel, find $\mathbf{a}\times \mathbf{b}$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1761382794354-dlh68tm1h" data-list-item-path="item-1761382794354-dlh68tm1h"><div class="list-item-paragraph">If&nbsp;$\mathbf{a}$ and $\mathbf{b}$ are non-parallel, interpret geometrically the following vector equations:</div><ol class="i-list"><li data-list-item-id="subitem-1761382803879-5an8eoos2" data-list-item-path="item-1761382794354-dlh68tm1h/subitem-1761382803879-5an8eoos2"><div class="list-item-paragraph">$\mathbf{r}\times \left( \mathbf{b}-\mathbf{a} \right)=\mathbf{0}$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1761382814409-qxi0jig85" data-list-item-path="item-1761382794354-dlh68tm1h/subitem-1761382814409-qxi0jig85"><div class="list-item-paragraph">$\mathbf{r}\cdot \left( \mathbf{b}-\mathbf{a} \right)=0$.<span class="inline-marks">[2]</span></div></li></ol></li></ol>
<p>It is now given that the coordinates of $A$ and $B$ are $\left( -1,2,2 \right)$ and $\left( 6,-2,-4 \right)$ respectively.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-1761382829866-apotqnwp4" data-list-item-path="item-1761382829866-apotqnwp4"><div class="list-item-paragraph">Use a vector product to find the area of the triangle $OAB$.<span class="inline-marks">[3]</span></div></li></ol>

Referred to the origin $O$, a variable point $R$ has position vector $\mathbf{r}$ and the fixed points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a}$ and $\mathbf{b}$ are non-zero vectors.

It is now given that the coordinates of $A$ and $B$ are $\left( -1,2,2 \right)$ and $\left( 6,-2,-4 \right)$ respectively.

Areas of Triangles and Parallelograms Using Vectors

2024 HCI Promo Q9

<p>Relative to the origin $O$, points $A$, $B$ and $C$ have position vectors $\mathbf{a}$, $\mathbf{b}$ and $3\mathbf{a}-4\mathbf{b}$ respectively, where&nbsp;$\mathbf{a}$ and $\mathbf{b}$ are non-zero and non-parallel.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Given that&nbsp;$OAB$ is an equilateral triangle, show that&nbsp;$\mathbf{a}\cdot \mathbf{b}=k{{\left| \mathbf{a} \right|}^{2}}$, where&nbsp;$k$ is a constant to be determined.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Deduce whether $OA$ and $AC$ are perpendicular.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Interpret geometrically the vector equation $\mathbf{r}\times \left( \mathbf{b}-\mathbf{a} \right)=\mathbf{a}\times \left( \mathbf{b}-\mathbf{a} \right)$.<span class="inline-marks">[2]</span></div></li></ol>

Relative to the origin $O$, points $A$, $B$ and $C$ have position vectors $\mathbf{a}$, $\mathbf{b}$ and $3\mathbf{a}-4\mathbf{b}$ respectively, where $\mathbf{a}$ and $\mathbf{b}$ are non-zero and non-parallel.

Scalar (Dot) Product

2022 NJC P1 Q3

<p>Two distinct points $A$ and $B$ have non-zero position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. The point $R$, with position vector $\mathbf{r}$, varies such that&nbsp;$\mathbf{r}\times \mathbf{a}=\mathbf{b}\times \mathbf{a}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;$\mathbf{r}-\mathbf{b}=k\,\mathbf{a}$ where&nbsp;$k$ is a real number.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Describe geometrically the set of all possible positions of the point $R$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given further that&nbsp;$\angle AOB={{90}^{\circ }}$, show that&nbsp;$\left| \mathbf{r}\cdot \mathbf{\hat{b}} \right|=\left| \mathbf{b} \right|$.<span class="inline-marks">[4]</span></div></li></ol>

Two distinct points $A$ and $B$ have non-zero position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. The point $R$, with position vector $\mathbf{r}$, varies such that $\mathbf{r}\times \mathbf{a}=\mathbf{b}\times \mathbf{a}$.

2022 SAJC Promo Q9

<p>The points $P$, $Q$ and $R$ have position vectors $\mathbf{p}$, $\mathbf{q}$ and $\mathbf{r}$ respectively, where&nbsp;$\mathbf{p}$ and $\mathbf{q}$ are non-zero and non-parallel. The points $P$ and $Q$ are fixed and $R$ varies.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Given that&nbsp;$\mathbf{r}\times \mathbf{q}=-\mathbf{q}\times \mathbf{p}$, describe geometrically the set of all possible positions of the point $R$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Given instead that&nbsp;$\mathbf{r}=\left( \begin{matrix}
x \\
y \\
z \\
\end{matrix} \right)$, $\mathbf{p}=\left( \begin{matrix}
1 \\
-2 \\
3 \\
\end{matrix} \right)$, $\mathbf{q}=\left( \begin{matrix}
{{q}_{1}} \\
{{q}_{2}} \\
{{q}_{3}} \\
\end{matrix} \right)$, and that&nbsp;$\mathbf{q}\cdot \left( \mathbf{p}-\mathbf{r} \right)=0$, find the relationship between $x$, $y$ and $z$ in terms of ${{q}_{1}}$, ${{q}_{2}}$ and ${{q}_{3}}$. Describe the set of all possible positions of the point $R$ in this case.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">It is now given that&nbsp;$\left| \mathbf{q} \right|=1$ and $C$ is a point with position vector $\mathbf{c}$ such that&nbsp;$\mathbf{q}\cdot \left( \mathbf{p}-\mathbf{c} \right)\ne 0$. Give a geometrical meaning of $\left| \mathbf{q}\cdot \left( \mathbf{p}-\mathbf{c} \right) \right|$.<span class="inline-marks">[1]</span></div></li></ol>

The points $P$, $Q$ and $R$ have position vectors $\mathbf{p}$, $\mathbf{q}$ and $\mathbf{r}$ respectively, where $\mathbf{p}$ and $\mathbf{q}$ are non-zero and non-parallel. The points $P$ and $Q$ are fixed and $R$ varies.

2022 YIJC P1 Q4

<ol class="abc-list"><li><div class="list-item-paragraph">The non-zero vectors $\mathbf{a,b}$ and $\mathbf{c}$ are such that $\mathbf{a}\times \mathbf{b}=\mathbf{c}\times \mathbf{a}$. Given that $\mathbf{b}\ne -\mathbf{c}$, find a linear relationship between $\mathbf{a,b}$ and $\mathbf{c}$.</div></li><li><div class="list-item-paragraph">The variable vector $\mathbf{v}$ satisfies the equation $\mathbf{v}\times (\mathbf{i}-3\mathbf{k})=2\mathbf{j}$. Find the set of vectors $\mathbf{v}$ and fully describe this set geometrically.</div></li></ol>

NYJC 4B Tutorial Q10

<ol class="abc-list"><li><div class="list-item-paragraph">Three distinct points $O$, $A$ and $B$ are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$.</div><ol class="i-list"><li><div class="list-item-paragraph">Let $\overrightarrow{OP}=\mathbf{r}$ and $\mathbf{r}\times \mathbf{b}=\mathbf{a}\times \mathbf{b}$. By expressing $\mathbf{r}$ in terms of $\mathbf{a}$ and $\mathbf{b}$, show that the locus of $P$ is a line parallel to $\mathbf{b}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">In terms of $\mathbf{a}$ and $\mathbf{b}$, write down the distance from $O$ to this line.</div><div class="list-item-paragraph">Hence write down the distance from $B$ to this line.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Given that the line passes through $O$, what can you say about the relationship between $\mathbf{a}$ and $\mathbf{b}$?</div><p style="text-align: right;">[1]</p></li></ol></li></ol>

Distance from a Point to a Line

2016 HCI J2 CT P1 Q10

<p>With reference to an origin $O$, a particle $P$ moves in space with position vector $\left( \lambda -\mu  \right)\mathbf{i}+\left( 1+2\mu  \right)\mathbf{j}+\left( 2-3\lambda  \right)\mathbf{k}$. Another particle $Q$ moves along the line $l$ with equation $\mathbf{r}=\left( \begin{matrix}
   1  \\
   1  \\
   0  \\
\end{matrix} \right)+t\left( \begin{matrix}
   2  \\
   -2  \\
   -3  \\
\end{matrix} \right)$, $t\in \mathbb{R}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">State the locus of $P$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Determine if the particles $P$ and $Q$ can meet.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the shortest possible distance between $P$ and $Q$.<span class="inline-marks">[2]</span></div></li></ol>
<p>Another particle $R$ moves along the line $m$ with equation $\mathbf{r}=\left( \begin{matrix}
   1  \\
   k  \\
   0  \\
\end{matrix} \right)+s\left( \begin{matrix}
   2  \\
   -2  \\
   -3k  \\
\end{matrix} \right)$,$s\in \mathbb{R}$, where&nbsp;$k$ is a constant.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Find condition(s) satisfied by $k$ if lines $l$ and $m$ are skew lines.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">A particle is shot from $X\left( 0,-1,-5 \right)$ perpendicularly toward the path of $Q$. Find the coordinates of the point where it crosses the path of $Q$.<span class="inline-marks">[2]</span></div></li></ol>

With reference to an origin $O$, a particle $P$ moves in space with position vector $\left( \lambda -\mu  \right)\mathbf{i}+\left( 1+2\mu  \right)\mathbf{j}+\left( 2-3\lambda  \right)\mathbf{k}$. Another particle $Q$ moves along the line $l$ with equation $\mathbf{r}=\left( \begin{matrix}
   1  \\
   1  \\
   0  \\
\end{matrix} \right)+t\left( \begin{matrix}
   2  \\
   -2  \\
   -3  \\
\end{matrix} \right)$, $t\in \mathbb{R}$.

Another particle $R$ moves along the line $m$ with equation $\mathbf{r}=\left( \begin{matrix}
   1  \\
   k  \\
   0  \\
\end{matrix} \right)+s\left( \begin{matrix}
   2  \\
   -2  \\
   -3k  \\
\end{matrix} \right)$,$s\in \mathbb{R}$, where $k$ is a constant.

Relationships Between a Line and a Plane

Relationships Between Two Lines

2017 NYJC P1 Q9

<ol class="abc-list"><li><div class="list-item-paragraph">Referred to the origin $O$, the fixed points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a}$ and $\mathbf{b}$ are non-zero vectors. It is given that $O$ divides the line segment $AC$ in the ratio $1:3$. Point $D$ divides the line segment $AB$ in the ratio $\lambda :1-\lambda $. Point $E$ divides the line segment $BC$ in the ratio $\mu :1-\mu $.</div><div class="list-item-paragraph">Show that the area of triangle $ODE$ can be written as $k\left| \mathbf{a}\times \mathbf{b} \right|$, where $k$ is a constant to be found in terms of $\mu $ and $\lambda $.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">A variable point $S$ has position vector $\mathbf{s}$ relative to the origin $O$. Given that $\mathbf{s}\times \mathbf{b}=-\mathbf{b}\times \mathbf{a}$, describe geometrically the set of all possible positions of the point $S$.</div><p style="text-align: right;">[3]</p></li></ol>

2022 CJC P2 Q4

<p>Relative to the origin, the fixed point $A$ has position vector $\mathbf{a}$. The line $l$ has equation $\mathbf{r}=s\mathbf{d}$, where&nbsp;$s$ is a parameter and $\mathbf{d}$ is a unit vector not parallel to $\mathbf{a}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Interpret geometrically the vector equation $\mathbf{r}=\mathbf{a}+t\,\mathbf{d}$, $t\in \mathbb{R}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find, in terms of $\mathbf{a}$ and $\mathbf{d}$, the position vector of the foot of perpendicular from $A$ to $l$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">A variable point $P$ with position vector $\mathbf{p}$ lies on the plane containing $A$ and $l$. Describe the locus of $P$ if&nbsp;$\left| \mathbf{p}\times \mathbf{d} \right|=\left| \mathbf{a}\times \mathbf{d} \right|$.<span class="inline-marks">[2]</span></div></li></ol>

Relative to the origin, the fixed point $A$ has position vector $\mathbf{a}$. The line $l$ has equation $\mathbf{r}=s\mathbf{d}$, where $s$ is a parameter and $\mathbf{d}$ is a unit vector not parallel to $\mathbf{a}$.

2023 NJC P2 Q1

<ol class="i-list"><li><div class="list-item-paragraph">Interpret geometrically the vector equation $\mathbf{r}=\mathbf{a}+t\mathbf{b}$, where $\mathbf{a}$ and $\mathbf{b}$ are constant vectors and $t$ is a parameter.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Interpret geometrically the vector equation $\mathbf{r}\cdot \mathbf{n}=d$, where $\mathbf{n}$ is a constant unit vector and $d$ is a constant scalar, stating what $d$ represents.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Given that $\mathbf{b}\cdot \mathbf{n}\ne 0$, solve the equations $\mathbf{r}=\mathbf{a}+t\mathbf{b}$ and $\mathbf{r}\cdot \mathbf{n}=d$ to find $\mathbf{r}$ in terms of $\mathbf{a}$, $\mathbf{b}$, $\mathbf{n}$ and $d$. Interpret the solution geometrically.</div><p style="text-align: right;">[3]</p></li></ol>

Intersection of a Line and a Plane

N2017 P1 Q6

<ol class="i-list"><li><div class="list-item-paragraph">The non-zero vectors $\mathbf{a}$,$\mathbf{b}$ and $\mathbf{c}$ are such that $\mathbf{a}\times \mathbf{b}=\mathbf{c}\times \mathbf{a}$. Given that $\mathbf{b}\ne -\mathbf{c}$, find a linear relationship between $\mathbf{a}$,$\mathbf{b}$ and $\mathbf{c}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">The variable vector $\mathbf{v}$ satisfies the equation $\mathbf{v}\times \left( \mathbf{i}-3\mathbf{k} \right)=2\mathbf{j}$. Find the set of vectors $\mathbf{v}$ and describe this set geometrically.</div><p style="text-align: right;">[5]</p></li></ol>

2017 Specimen Paper P1 Q6

<ol class="abc-list"><li><ol class="i-list"><li><div class="list-item-paragraph">Find the shortest distance from the origin $O$ to the plane $p$ with equation $2x+3y+6z=10$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given that point $A$ lies on plane $p$ and $OA=10$, find the acute angle between the plane $p$ and $OA$.<span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-paragraph">Two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ are such that&nbsp;$\left| \mathbf{a}\times \mathbf{b} \right|=\sqrt{3}$, $\left| \mathbf{a}-\mathbf{b} \right|=1$ and the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ is ${{30}^{\circ }}$, find the possible values of $\left| \mathbf{a} \right|$ and $\left| \mathbf{b} \right|$.<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">Two fixed points $C$ and $D$ have position vectors $\mathbf{c}$ and $\mathbf{d}$ respectively. By considering the result $\mathbf{c}\times \mathbf{c}=\mathbf{0}$, give a geometrical interpretation of the vector equation $\mathbf{r}\times \left( \mathbf{d}-\mathbf{c} \right)=\mathbf{c}\times \mathbf{d}$.<span class="inline-marks">[3]</span></div></li></ol>

Magnitude of a Vector

2023 NJC Promo Q5

<ol class="abc-list"><li><div class="list-item-paragraph">Relative to the origin $O$, the fixed points $A$ and $B$ are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$. The point $R$ has position vector $\overrightarrow{OR}=\left( \begin{matrix}
x \\
y \\
z \\
\end{matrix} \right)$. Interpret geometrically the point $R$, stating what $d$ represents, given that $\overrightarrow{OR}\cdot \frac{\left( \mathbf{a}\times \mathbf{b} \right)}{\left| \mathbf{a}\times \mathbf{b} \right|}=d$ where $d$ is a constant scalar.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-image"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761292784879-bx46kcj7p.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEyOTI3ODQ4NzktYng0NmtjajdwLnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3OTM1MDQ3NzYsImV4cCI6MTc5MzUwNTA3Nn0.FK5vC0fCe6H7s_jEwrBSN6D9ToResgHrR2eTcevp8Qc" alt="List item image" /></div><ol class="i-list"><li><div class="list-item-paragraph">The points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively relative to the origin $O$. $M$ is the mid- point of $AB$. The point $N$ on $OB$ is such that $ON:NB=k:1-k$, where $0 < k < 1$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">The lines $OM$ and $AN$ intersect at the point $P$. Given that $OP:PM=2k:1-k$, show that $\overrightarrow{OP}=\frac{k}{1+k}\left( \mathbf{a}+\mathbf{b} \right)$.</div><div class="list-item-paragraph">Hence, find the area of triangle $OPN$ in terms of $k$.</div><p style="text-align: right;">[4]</p></li></ol></li></ol>

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