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H2 Math

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N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>Relative to the origin $O$, the position vectors of $A$, $B$ and $C$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively where&nbsp;$\mathbf{c}=3\mathbf{b}-\mathbf{a}$. Given that&nbsp;$\mathbf{b}$ is a unit vector, $|\mathbf{a}|=3$, $|\mathbf{b}-\mathbf{a}|=\sqrt7$ and the angle between $\mathbf{a}$ and $\mathbf{b}$ is $\frac\pi3$,</p>
<ol class="i-list"><li data-list-item-id="asrjc-q06-item01" data-list-item-path="asrjc-q06-item01"><div class="list-item-paragraph">find the exact shortest distance from $C$ to the line $AB$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="asrjc-q06-item02" data-list-item-path="asrjc-q06-item02"><div class="list-item-paragraph">Find the angle between $\mathbf{b}$ and $\overrightarrow{AC}$.<span class="inline-marks">[3]</span></div></li></ol>

Relative to the origin $O$, the position vectors of $A$, $B$ and $C$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively where $\mathbf{c}=3\mathbf{b}-\mathbf{a}$. Given that $\mathbf{b}$ is a unit vector, $|\mathbf{a}|=3$, $|\mathbf{b}-\mathbf{a}|=\sqrt7$ and the angle between $\mathbf{a}$ and $\mathbf{b}$ is $\frac\pi3$,

find the exact shortest distance from $C$ to the line $AB$.

Find the angle between $\mathbf{b}$ and $\overrightarrow{AC}$.

Angles Between Vectors

Distance from a Point to a Line

2026 ASRJC P1 Q6

Vectors

<p>Relative to the origin $O$, two points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. It is given that&nbsp;$|\mathbf{a}|=2$, $|\mathbf{b}|=1$ and $|3\mathbf{a}-2\mathbf{b}|=\sqrt{37}$.</p>
<ol class="i-list"><li data-list-item-id="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-0" data-list-item-path="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-0"><div class="list-item-paragraph">By considering the scalar product $(3\mathbf{a}-2\mathbf{b})\cdot(3\mathbf{a}-2\mathbf{b})$, show that&nbsp;$\mathbf{a}\cdot\mathbf{b}=\dfrac{1}{4}$, and give the geometrical meaning of $|\mathbf{a}\cdot\mathbf{b}|$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-1" data-list-item-path="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-1"><div class="list-item-paragraph">By finding the cosine of the angle between vectors $\mathbf{a}$ and $\mathbf{b}$, deduce the exact value of $|(\mathbf{a}-\mathbf{b})\times\mathbf{b}|$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-2" data-list-item-path="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-2"><div class="list-item-paragraph">Interpret geometrically the value of $|(\mathbf{a}-\mathbf{b})\times\mathbf{b}|$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-3" data-list-item-path="h2-2025-sajc-promo-q8-sajc-q8-item-block-1-item-3"><div class="list-item-paragraph">Write down, in terms of $\mathbf{a}$ and $\mathbf{b}$, a vector equation of the line that passes through $O$ and bisects the angle $AOB$.<span class="inline-marks">[2]</span></div></li></ol>

Relative to the origin $O$, two points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. It is given that $|\mathbf{a}|=2$, $|\mathbf{b}|=1$ and $|3\mathbf{a}-2\mathbf{b}|=\sqrt{37}$.

By considering the scalar product $(3\mathbf{a}-2\mathbf{b})\cdot(3\mathbf{a}-2\mathbf{b})$, show that $\mathbf{a}\cdot\mathbf{b}=\dfrac{1}{4}$, and give the geometrical meaning of $|\mathbf{a}\cdot\mathbf{b}|$.

By finding the cosine of the angle between vectors $\mathbf{a}$ and $\mathbf{b}$, deduce the exact value of $|(\mathbf{a}-\mathbf{b})\times\mathbf{b}|$.

Interpret geometrically the value of $|(\mathbf{a}-\mathbf{b})\times\mathbf{b}|$.

Write down, in terms of $\mathbf{a}$ and $\mathbf{b}$, a vector equation of the line that passes through $O$ and bisects the angle $AOB$.

Projection Lengths

Angle Bisectors Using Vectors

Equations of Lines

Vector (Cross) Product

Areas of Triangles and Parallelograms Using Vectors

Scalar (Dot) Product

2025 SAJC Promo Q8

<p>Referred to the origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ such that</p>
<p style="text-align: center">\[\mathbf{a} = \mathbf{i} + \mathbf{j} + \mathbf{k} \quad \text{and} \quad \mathbf{b} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k} .\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the size of angle $OAB$.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>The point $C$ has position vector $\mathbf{c}$ given by $\mathbf{c} = \lambda \mathbf{a} + \mu \mathbf{b}$, where&nbsp;$\lambda$ and $\mu$ are positive constants.<br>Given that the area of triangle $OAC$ is twice that of triangle $OBC$,</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">find $\mu$ in terms of $\lambda$,</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">hence, if&nbsp;$OC = \sqrt{118}$, find the position vector $\mathbf{c}$.</span><span class="inline-marks">[4]</span></div></li></ol>

Referred to the origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ such that

\[\mathbf{a} = \mathbf{i} + \mathbf{j} + \mathbf{k} \quad \text{and} \quad \mathbf{b} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k} .\]

The point $C$ has position vector $\mathbf{c}$ given by $\mathbf{c} = \lambda \mathbf{a} + \mu \mathbf{b}$, where $\lambda$ and $\mu$ are positive constants.
Given that the area of triangle $OAC$ is twice that of triangle $OBC$,

hence, if $OC = \sqrt{118}$, find the position vector $\mathbf{c}$.

Magnitude of a Vector

2021 JPJC P1 Q2

<p>With respect to an origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where&nbsp;$\mathbf{a}$ and $\mathbf{b}$ are not parallel.</p>
<ol class="i-list"><li><div class="list-item-paragraph">It is given that&nbsp;$B$ lies on the line segment $AC$, such that&nbsp;$\overrightarrow{BC}=2\mathbf{b}-k\mathbf{a}$. State the value of the constant $k$. Hence, 	find $\overrightarrow{OC}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">$\overrightarrow{ON}$ is the projection vectors of $\overrightarrow{OB}$ onto $\overrightarrow{OA}$, where&nbsp;$\overrightarrow{ON}=\frac{3}{4}\mathbf{a}$ and $\mathbf{a}\cdot \mathbf{b}=4$. Show that the length of $OA$ is $\frac{4\sqrt{3}}{3}$.	<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given also that&nbsp;$\mathbf{b}=\left( \begin{matrix}
   2  \\
   -2  \\
   2  \\
\end{matrix} \right)$, find the angle between $\overrightarrow{OA}$ and  $\overrightarrow{OB}$.	<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Given that&nbsp;$OADC$ is a parallelogram and $E$ lies on line segment $AC$ such that&nbsp;$AE:AC=2:3$, find $\overrightarrow{DE}$.<span class="inline-marks">[3]</span></div></li></ol>

With respect to an origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a}$ and $\mathbf{b}$ are not parallel.

Vector Projections

Collinearity Using Vectors

Vectors in a Parallelogram

Ratio Theorem and Section Formula

<p>Given&nbsp;$\left| \mathbf{a} \right|=7$ and $\left| \mathbf{b} \right|=4$, $\mathbf{a}\cdot \mathbf{b}=8$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Calculate the angle between vectors $\mathbf{a}$ and $\mathbf{b}$.</div></li><li><div class="list-item-paragraph">Find the values of $(\mathbf{a}+\mathbf{b})\cdot (\mathbf{a}+\mathbf{b})$ and $(\mathbf{a}-\mathbf{b})\cdot (\mathbf{a}-\mathbf{b})$.</div></li><li><div class="list-item-paragraph">Hence, find the values of $\left| \mathbf{a}+\mathbf{b} \right|$ and $\left| \mathbf{a}-\mathbf{b} \right|$.</div></li></ol>

Given $\left| \mathbf{a} \right|=7$ and $\left| \mathbf{b} \right|=4$, $\mathbf{a}\cdot \mathbf{b}=8$.

<p>The acute angle between two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ is $\alpha$ radians, and the vectors $\mathbf{b}$ and $2\mathbf{a}-\mathbf{b}$ are perpendicular. Find the value of $\alpha$ if&nbsp;$|\mathbf{b}| = \sqrt{2}|\mathbf{a}|$.<span class="inline-marks">[3]</span></p>

The acute angle between two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ is $\alpha$ radians, and the vectors $\mathbf{b}$ and $2\mathbf{a}-\mathbf{b}$ are perpendicular. Find the value of $\alpha$ if $|\mathbf{b}| = \sqrt{2}|\mathbf{a}|$.

Perpendicular Vectors Using the Scalar Product

2025 YIJC P2 Q1

<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/1768703533158-y2y9mw7e2d.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg3MDM1MzMxNTgteTJ5OW13N2UyZC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0ODc1LCJleHAiOjE3OTM1MDUxNzV9.rYIOO6Uldm569uTAY_3dXjk3PoMffIjabmJSaJDArTs" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Relative to the origin $O$, the points $A$, $B$, and $C$ have position vectors $\mathbf{a}$, $\mathbf{b}$ and $3\mathbf{a}+\mathbf{b}$ respectively. The point $D$ lies on $AB$ such that&nbsp;$AD=kAB$, where&nbsp;$0&lt;k&lt;1$.</p>
<ol class="abc-list"><li data-list-item-id="item-6-a"><div class="list-item-content"><span class="list-item-text">Find a vector equation of the line $OD$ in terms of $k$, $\mathbf{a}$ and $\mathbf{b}$.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-6-b"><div class="list-item-content"><span class="list-item-text">The point $E$ is the midpoint of $BC$. Find the value of $k$ if&nbsp;$O$, $D$ and $E$ are collinear.</span><span class="inline-marks">[4]</span></div></li></ol>
<p>It is given that&nbsp;$|\mathbf{a}|=1$, $|\mathbf{b}|=2$ and $|3\mathbf{a}-2\mathbf{b}|=\sqrt{31}$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-6-c"><div class="list-item-content"><span class="list-item-text">By considering the scalar product $(3\mathbf{a}-2\mathbf{b}) \cdot (3\mathbf{a}-2\mathbf{b})$, find the numerical value of $\mathbf{a} \cdot \mathbf{b}$ and hence determine the angle between $\mathbf{a}$ and $\mathbf{b}$.</span><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-6-d"><div class="list-item-content"><span class="list-item-text">Give a geometrical interpretation of $|\mathbf{a} \cdot \mathbf{b}|$.</span><span class="inline-marks">[1]</span></div></li></ol>

Relative to the origin $O$, the points $A$, $B$, and $C$ have position vectors $\mathbf{a}$, $\mathbf{b}$ and $3\mathbf{a}+\mathbf{b}$ respectively. The point $D$ lies on $AB$ such that $AD=kAB$, where $0<k<1$.

Find a vector equation of the line $OD$ in terms of $k$, $\mathbf{a}$ and $\mathbf{b}$.

The point $E$ is the midpoint of $BC$. Find the value of $k$ if $O$, $D$ and $E$ are collinear.

It is given that $|\mathbf{a}|=1$, $|\mathbf{b}|=2$ and $|3\mathbf{a}-2\mathbf{b}|=\sqrt{31}$.

By considering the scalar product $(3\mathbf{a}-2\mathbf{b}) \cdot (3\mathbf{a}-2\mathbf{b})$, find the numerical value of $\mathbf{a} \cdot \mathbf{b}$ and hence determine the angle between $\mathbf{a}$ and $\mathbf{b}$.

Give a geometrical interpretation of $|\mathbf{a} \cdot \mathbf{b}|$.

2025 DHS P2 Q4

<p>Given two vectors $\mathbf{a}=2\mathbf{i}+\mathbf{j}-2\mathbf{k}$ and $\mathbf{b}=\mathbf{j}+3\mathbf{k}$, find:</p>
<ol class="i-list"><li><div class="list-item-paragraph">a unit vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$.</div></li><li><div class="list-item-paragraph">the exact value of $\sin \theta $, where $\theta $ is the angle between $\mathbf{a}$ and $\mathbf{b}$.</div></li></ol>

Given two vectors $\mathbf{a}=2\mathbf{i}+\mathbf{j}-2\mathbf{k}$ and $\mathbf{b}=\mathbf{j}+3\mathbf{k}$, find:

Unit Vectors

Unit 13.1 Worked Example 4

<p>Find the acute angle between direction vectors $\left( -6\mathbf{i}+\mathbf{j} \right)$ and $\left( \mathbf{i}-\mathbf{j}-8\mathbf{k} \right)$.</p>

Find the acute angle between direction vectors $\left( -6\mathbf{i}+\mathbf{j} \right)$ and $\left( \mathbf{i}-\mathbf{j}-8\mathbf{k} \right)$.

Unit 13.1 Worked Example 3

<p>Given two vectors $\mathbf{a}=2\mathbf{i}+\mathbf{j}-2\mathbf{k}$ and $\mathbf{b}=3\mathbf{i}+4\mathbf{j}-\mathbf{k}$, find:</p>
<ol class="i-list"><li><div class="list-item-paragraph">a unit vector perpendicular to both&nbsp;$\mathbf{a}$ and $\mathbf{b}$.</div></li><li><div class="list-item-paragraph">the exact value of $\sin\theta$, where&nbsp;$\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$.</div></li></ol>

Given two vectors $\mathbf{a}=2\mathbf{i}+\mathbf{j}-2\mathbf{k}$ and $\mathbf{b}=3\mathbf{i}+4\mathbf{j}-\mathbf{k}$, find:

<p>Given two vectors $\mathbf{a}=2\mathbf{i}+\mathbf{j}-2\mathbf{k}$ and $\mathbf{b}=\mathbf{j}+3\mathbf{k}$, find:</p>
<ol class="i-list"><li><div class="list-item-paragraph">a unit vector perpendicular to both&nbsp;$\mathbf{a}$ and $\mathbf{b}$.</div></li><li><div class="list-item-paragraph">the exact value of $\sin\theta$, where&nbsp;$\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$.</div></li></ol>

<ol class="abc-list"><li><div class="list-item-paragraph">Evaluate $\left( 2\mathbf{i}+3\mathbf{j}-\mathbf{k} \right)\cdot \left( \mathbf{i}-5\mathbf{k} \right)$.</div></li><li><div class="list-item-paragraph">Evaluate $\left( \mathbf{i}-5\mathbf{j}+4\mathbf{k} \right)\cdot \left( 3\mathbf{i}-2\mathbf{j}-3\mathbf{k} \right)$.</div></li><li><div class="list-item-paragraph">Evaluate $\left( -6\mathbf{i}+\mathbf{j} \right)\cdot \left( \mathbf{i}-\mathbf{j}-8\mathbf{k} \right)$.</div></li><li><div class="list-item-paragraph">Find the acute angle between direction vectors $\left( 2\mathbf{i}+3\mathbf{j}-\mathbf{k} \right)$ and $\left( \mathbf{i}-5\mathbf{k} \right)$.</div></li><li><div class="list-item-paragraph">Find the acute angle between direction vectors $\left( \mathbf{i}-5\mathbf{j}+4\mathbf{k} \right)$ and $\left( 3\mathbf{i}-2\mathbf{j}-3\mathbf{k} \right)$.</div></li><li><div class="list-item-paragraph">Find the acute angle between direction vectors $\left( -6\mathbf{i}+\mathbf{j} \right)$ and $\left( \mathbf{i}-\mathbf{j}-8\mathbf{k} \right)$.</div></li></ol>

<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1761382105821-2y2g8mc3ico.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzODIxMDU4MjEtMnkyZzhtYzNpY28ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDg3NSwiZXhwIjoxNzkzNTA1MTc1fQ.w5JGOigaKgq1drcMOGqyMs_ROerYqDxDnIqW_eURmZQ" width="400" class="aligncenter" /></figure>
<p>Methane $\left( \mathrm{C}{{\mathrm{H}}_{4}} \right)$ is a chemical compound with a tetrahedral structure. The $4$ hydrogen $\left( \mathrm{H} \right)$ atoms form a $4$ regular tetrahedron, and the carbon $\left( \mathrm{C} \right)$ atom is in the centre.</p>
<p>Let the centre of the $\mathrm{C}$-atom be the point $P$, and the centres of the $4$ $\mathrm{H}$-atoms be the points $Q$, $R$, $S$ and $T$. The coordinates of $P$, $Q$, $R$ and $S$ are $\left( -1,\,\,0,\,\,2 \right)$, $\left( -2,\,\,-1,\,\,1 \right)$, $\left( -2,\,\,1,\,\,3 \right)$ and $\left( 0,\,\,-1,\,\,\alpha \right)$ respectively.</p>
<p>The angle $\theta $ subtended by any two $\mathrm{C-H}$bonds at the $\mathrm{C}$-atom, such as angle $QPR$, is known as the $\mathrm{H-C-H}$ bond angle (see diagram above).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the bond angle, correct to $2$ decimal places.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">By using the fact that $QS=RS$, show that $a=3$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find a cartesian equation for plane $\pi $, which contains the points $P$, $Q$ and $R$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">$F$ is the point on $\pi $ that is closest to the point $S$.</div><ol class="i-list"><li><div class="list-item-paragraph">State a vector equation for the line $SF$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Hence, show that the coordinates of $F$ are $\left( 0,\,\,0,\,\,2 \right)$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Given that the point $T$ is the mirror image of the point $S$ in $\pi $, find the position vector of $T$.</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

Methane $\left( \mathrm{C}{{\mathrm{H}}_{4}} \right)$ is a chemical compound with a tetrahedral structure. The $4$ hydrogen $\left( \mathrm{H} \right)$ atoms form a $4$ regular tetrahedron, and the carbon $\left( \mathrm{C} \right)$ atom is in the centre.

Let the centre of the $\mathrm{C}$-atom be the point $P$, and the centres of the $4$ $\mathrm{H}$-atoms be the points $Q$, $R$, $S$ and $T$. The coordinates of $P$, $Q$, $R$ and $S$ are $\left( -1,\,\,0,\,\,2 \right)$, $\left( -2,\,\,-1,\,\,1 \right)$, $\left( -2,\,\,1,\,\,3 \right)$ and $\left( 0,\,\,-1,\,\,\alpha \right)$ respectively.

The angle $\theta $ subtended by any two $\mathrm{C-H}$bonds at the $\mathrm{C}$-atom, such as angle $QPR$, is known as the $\mathrm{H-C-H}$ bond angle (see diagram above).

Perpendicular Projection and Reflection in a Plane

Three-Dimensional Geometry

Equations of Planes

2023 EJC Promo Q12

<p>Relative to the origin $O$, two points $A$ and $B$ have position vectors given by $\mathbf{a}=2\mathbf{i}-\mathbf{j}$ and $\mathbf{b}=\alpha \mathbf{k}$ respectively, where&nbsp;$\alpha $ is a negative real constant.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the angle between $OA$ and $OB$.<span class="inline-marks">[1]</span></div></li></ol>
<p>Let&nbsp;$\angle OBA=\theta $, where&nbsp;$\theta $ is in degrees. It is given that&nbsp;$\sin \theta =\frac{1}{2}$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Find the two possible values of $\theta $, and justify which of the two values is the correct value of $\theta $.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">By considering a suitable scalar product, find the exact value of $\alpha $.<span class="inline-marks">[3]</span></div></li></ol>
<p>A point $C$ has position vector $\mathbf{c}$ such that&nbsp;$\mathbf{c}=\mathbf{b}-\mathbf{a}$.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">State the shape of the quadrilateral $OABC$, justifying your answer. Find the exact area of the quadrilateral $OABC$.<span class="inline-marks">[3]</span></div></li></ol>

Relative to the origin $O$, two points $A$ and $B$ have position vectors given by $\mathbf{a}=2\mathbf{i}-\mathbf{j}$ and $\mathbf{b}=\alpha \mathbf{k}$ respectively, where $\alpha $ is a negative real constant.

Let $\angle OBA=\theta $, where $\theta $ is in degrees. It is given that $\sin \theta =\frac{1}{2}$.

A point $C$ has position vector $\mathbf{c}$ such that $\mathbf{c}=\mathbf{b}-\mathbf{a}$.

2023 HCI P1 Q8

<p>With reference to the origin $O$, the points $A$ and $B$ have coordinates $\left( 1,2,0 \right)$ and $\left( -5,1,0 \right)$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the angle $AOB$.<span class="inline-marks">[3]</span></div></li></ol>
<p>The vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ are projected along the direction of $\left( \begin{matrix}
1 \\
2 \\
2 \\
\end{matrix} \right)$ and $\left( \begin{matrix}
2 \\
1 \\
2 \\
\end{matrix} \right)$ to form the vectors $\overrightarrow{O'A'}$ and $\overrightarrow{O'B'}$.<span class="inline-marks">[3]</span></p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Find the vectors $\overrightarrow{O'A'}$ and $\overrightarrow{O'B'}$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the exact area of the triangle $O'A'B'$.<span class="inline-marks">[2]</span></div></li></ol>

With reference to the origin $O$, the points $A$ and $B$ have coordinates $\left( 1,2,0 \right)$ and $\left( -5,1,0 \right)$.

The vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ are projected along the direction of $\left( \begin{matrix}
1 \\
2 \\
2 \\
\end{matrix} \right)$ and $\left( \begin{matrix}
2 \\
1 \\
2 \\
\end{matrix} \right)$ to form the vectors $\overrightarrow{O'A'}$ and $\overrightarrow{O'B'}$.

2023 DHS Promo Q5

<p>Vectors $\mathbf{a}$ and $\mathbf{b}$ are such that&nbsp;$\mathbf{a}\cdot \mathbf{b}=-1$. It is also given that&nbsp;$\left( \mathbf{a}\times \mathbf{b}+\mathbf{a} \right)$ is perpendicular to $\left( \mathbf{a}\times \mathbf{b}+\mathbf{b} \right)$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;$\left| \mathbf{a}\times \mathbf{b} \right|=1$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence, find the angle between the direction of $\mathbf{a}$ and the direction of $\mathbf{b}$.<span class="inline-marks">[3]</span></div></li></ol>

Vectors $\mathbf{a}$ and $\mathbf{b}$ are such that $\mathbf{a}\cdot \mathbf{b}=-1$. It is also given that $\left( \mathbf{a}\times \mathbf{b}+\mathbf{a} \right)$ is perpendicular to $\left( \mathbf{a}\times \mathbf{b}+\mathbf{b} \right)$.

N2023 P1 Q3

<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/converted/1761366133661-p29dtcsjujq.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjEzNjYxMzM2NjEtcDI5ZHRjc2p1anEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDg3NSwiZXhwIjoxNzkzNTA1MTc1fQ.rRKTJ_scxbuGXGpTct-YQBOXn1ztR5yf-mUaYqGajCo" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Referred to the origin $O$, the points $P$ and $Q$ are such that&nbsp;$\overrightarrow{OP}=4\,\mathbf{a}$ and $\overrightarrow{OQ}=3\,\mathbf{b}$, where&nbsp;$\mathbf{a}$ and $\mathbf{b}$ are unit vectors. The points $M$ and $S$ are on $OP$ produced such that&nbsp;$M$ is the midpoint of $P$ and $S$. The point $R$ is on $OQ$ produced. (see diagram above)</p>
<p>It is given that&nbsp;$\overrightarrow{MQ}=3\mathbf{b}-7\mathbf{a}$, $\overrightarrow{MR}=\frac{15}{2}\mathbf{b}-7\mathbf{a}$ and $\left| \overrightarrow{MQ} \right|=\left| \overrightarrow{MR} \right|$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Using a suitable scalar product, show that&nbsp;$\cos \angle POQ=\frac{3}{4}$.<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">Find $\overrightarrow{SR}$ in terms of $\mathbf{a}$ and $\mathbf{b}$. Show that&nbsp;$\overrightarrow{SR}=\lambda \overrightarrow{PQ}$ where&nbsp;$\lambda $ is a constant to be found.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the exact area of the triangle $OPQ$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">By considering the ratio of the area of the triangle $OSR$ to the area of the triangle $OPQ$, determine the exact area of the quadrilateral $PQRS$.<span class="inline-marks">[2]</span></div></li></ol>

Referred to the origin $O$, the points $P$ and $Q$ are such that $\overrightarrow{OP}=4\,\mathbf{a}$ and $\overrightarrow{OQ}=3\,\mathbf{b}$, where $\mathbf{a}$ and $\mathbf{b}$ are unit vectors. The points $M$ and $S$ are on $OP$ produced such that $M$ is the midpoint of $P$ and $S$. The point $R$ is on $OQ$ produced. (see diagram above)

It is given that $\overrightarrow{MQ}=3\mathbf{b}-7\mathbf{a}$, $\overrightarrow{MR}=\frac{15}{2}\mathbf{b}-7\mathbf{a}$ and $\left| \overrightarrow{MQ} \right|=\left| \overrightarrow{MR} \right|$.

Parallel Vectors

Vector Subtraction

2022 TJC Promo Q10

<p>As part of a sculpture, an artist erects a flat triangular sheet $ABC$ in his garden. The vertices are attached to vertical poles $DA$, $EB$ and $FC$. The coordinate axes ${{O}_{x}}$ and ${{O}_{y}}$ are horizontal, and ${{O}_{z}}$ is vertical. The coordinates of triangle are $A(2,0,2)$, $B(-2,0,1)$ and $C(0,4,3)$, with unites in metres.</p>
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<ol class="i-list"><li data-list-item-id="item-1761364059851-9o95nyumb" data-list-item-path="item-1761364059851-9o95nyumb"><div class="list-item-paragraph">Find the length of the side $AC$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761364061861-aitdwj8ii" data-list-item-path="item-1761364061861-aitdwj8ii"><div class="list-item-paragraph">Find the scalar product of $\overrightarrow{AB}\cdot \overrightarrow{AC}$, and the angle $BAC$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1761364062813-cg59xpbtt" data-list-item-path="item-1761364062813-cg59xpbtt"><div class="list-item-paragraph">Show that&nbsp;$2\mathbf{i}+3\mathbf{j}-8\mathbf{k}$is perpendicular to the lines $AB$ and $AC$. Hence find the Cartesian equation of the plane $ABC$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1761364063698-hk7lenqyc" data-list-item-path="item-1761364063698-hk7lenqyc"><div class="list-item-paragraph">The artist decides to erect another vertical pole $GH$ based at the point $G(1,1,0)$. Calculate the height of the pole if&nbsp;$H$ is to lie in the plane $ABC$.<span class="inline-marks">[5]</span></div></li></ol>

As part of a sculpture, an artist erects a flat triangular sheet $ABC$ in his garden. The vertices are attached to vertical poles $DA$, $EB$ and $FC$. The coordinate axes ${{O}_{x}}$ and ${{O}_{y}}$ are horizontal, and ${{O}_{z}}$ is vertical. The coordinates of triangle are $A(2,0,2)$, $B(-2,0,1)$ and $C(0,4,3)$, with unites in metres.

Finding Points on a Plane

2007 TPJC P2 Q4

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-paragraph">The unit vector $\mathbf{d}$ makes angles of ${{60}^{\circ }}$ with both the $x$- and $y$-axes, and $\theta $ with the $z$-axis, where ${{0}^{\circ }}\le \theta \le {{90}^{\circ }}$.</div><div class="list-item-paragraph">Show that $\mathbf{d}$ is parallel to $\mathbf{i}+\mathbf{j}+\sqrt{2}\mathbf{k}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">The line $m$ is parallel to $\mathbf{d}$and passes through the point with coordinates $\left( 2,-1,0 \right)$. Find the coordinates of the point on $m$ that is closest to the point with coordinates $\left( 3,2,0 \right)$.</div><p style="text-align: right;">[3]</p></li></ol></li><li><div class="list-item-paragraph">The plane ${{p}_{1}}$ has equation $\mathbf{r}\cdot \left( \mathbf{i}+2\mathbf{j}+2\mathbf{k} \right)=5$, and the line $l$ has equation $\frac{x-a}{2}=\frac{y-1}{b}=-\frac{z}{2}$, where $a$ and $b$ are constants.</div><div class="list-item-paragraph">Given that $l$ lies on ${{p}_{1}}$, show that $b=1$ and find the value of $a$.</div><p style="text-align: right;">[2]</p><ol class="i-list"><li><div class="list-item-paragraph">The plane ${{p}_{2}}$ contains $l$ and is perpendicular to ${{p}_{1}}$. Find the equation of ${{p}_{2}}$ in the form $\mathbf{r}\cdot \mathbf{n}=c$, where $c$ is a constant to be determined.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">The variable point $P\left( x,y,z \right)$ is equidistant from ${{p}_{1}}$ and ${{p}_{2}}$. Find the cartesian equation(s) of the locus of $P$.</div><p style="text-align: right;">[3]</p></li></ol></li></ol>

Perpendicular Projection and Reflection in a Line

Distance from a Point to a Plane

Relationships Between a Line and a Plane

2017 ACJC P2 Q4

<p>The points $A$, $B$, $C$, $D$ have position vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, $\mathbf{d}$ given by $\mathbf{a}=\mathbf{i}+2\mathbf{j}+3\mathbf{k}$, $\mathbf{b}=\mathbf{i}+2\mathbf{j}+2\mathbf{k}$, $\mathbf{c}=3\mathbf{i}+2\mathbf{j}+\mathbf{k}$, $\mathbf{d}=4\mathbf{i}-\mathbf{j}-\mathbf{k}$, respectively. The point $P$ lies on $AB$ produced such that $AP=2AB$, and the point $Q$ is the midpoint of $AC$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $PQ$ is perpendicular to $AQ$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Find the area of the triangle $APQ$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find a vector perpendicular to the plane $ABC$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the cosine of the angle between $\overrightarrow{AD}$ and $\overrightarrow{BD}$.</div><p style="text-align: right;">[3]</p></li></ol>

The points $A$, $B$, $C$, $D$ have position vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, $\mathbf{d}$ given by $\mathbf{a}=\mathbf{i}+2\mathbf{j}+3\mathbf{k}$, $\mathbf{b}=\mathbf{i}+2\mathbf{j}+2\mathbf{k}$, $\mathbf{c}=3\mathbf{i}+2\mathbf{j}+\mathbf{k}$, $\mathbf{d}=4\mathbf{i}-\mathbf{j}-\mathbf{k}$, respectively. The point $P$ lies on $AB$ produced such that $AP=2AB$, and the point $Q$ is the midpoint of $AC$.

Math Question Bank