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N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>Let&nbsp;$\theta$ be the angle between the unit vectors $ \mathbf{a}$ and $ \mathbf{b}$, where&nbsp;$0 &lt; \theta &lt; \pi$. Express $\vert{}\mathbf{a} - \mathbf{b}\vert{}$ in terms of $\sin \frac{1}{2}\theta$</p>

Let $\theta$ be the angle between the unit vectors $ \mathbf{a}$ and $ \mathbf{b}$, where $0 < \theta < \pi$. Express $\vert{}\mathbf{a} - \mathbf{b}\vert{}$ in terms of $\sin \frac{1}{2}\theta$

Magnitude of a Vector

Scalar (Dot) Product

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

Angles Between Vectors

Angle Bisectors Using Vectors

Equations of Lines

Vector (Cross) Product

Areas of Triangles and Parallelograms Using Vectors

2025 SAJC Promo Q8

<ol class="abc-list"><li data-list-item-id="h2-2025-nyjc-promo-q7-nyjc-q7-item-block-0-item-0" data-list-item-path="h2-2025-nyjc-promo-q7-nyjc-q7-item-block-0-item-0"><div class="list-item-paragraph">Given that&nbsp;$\mathbf{a}$ and $\mathbf{b}$ are non-zero vectors such that&nbsp;$\mathbf{a}=(\mathbf{a}\cdot\mathbf{b})\mathbf{b}$. State the relationship between $\mathbf{a}$ and $\mathbf{b}$. Find $|\mathbf{b}|$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="h2-2025-nyjc-promo-q7-nyjc-q7-item-block-0-item-1" data-list-item-path="h2-2025-nyjc-promo-q7-nyjc-q7-item-block-0-item-1"><div class="list-item-paragraph">With reference to the origin $O$, the points $M$ and $N$ are such that&nbsp;$\overrightarrow{OM}=\alpha\mathbf{i}+4\mathbf{j}+\beta\mathbf{k}$ and $\overrightarrow{ON}=4\mathbf{i}+\alpha\mathbf{j}+\beta\mathbf{k}$, where&nbsp;$\alpha$ and $\beta$ are constants.</div><ol class="i-list"><li data-list-item-id="h2-2025-nyjc-promo-q7-nyjc-q7-item-block-0-item-1-sub-0" data-list-item-path="h2-2025-nyjc-promo-q7-nyjc-q7-item-block-0-item-1/h2-2025-nyjc-promo-q7-nyjc-q7-item-block-0-item-1-sub-0"><div class="list-item-paragraph">Given that&nbsp;$M$ lies in the $xy$-plane and $|\overrightarrow{OM}|=5$, find the values of $\alpha$ and $\beta$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-nyjc-promo-q7-nyjc-q7-item-block-0-item-1-sub-1" data-list-item-path="h2-2025-nyjc-promo-q7-nyjc-q7-item-block-0-item-1/h2-2025-nyjc-promo-q7-nyjc-q7-item-block-0-item-1-sub-1"><div class="list-item-paragraph"><p>Given <strong>instead</strong> that&nbsp;$\overrightarrow{OM}$ is perpendicular to $\overrightarrow{ON}$, find an expression for&nbsp;$|\overrightarrow{OM}\times\overrightarrow{ON}|$ in terms of $\alpha$, giving your answer in fully factorised form.</p><span class="inline-marks">[3]</span></div></li></ol></li></ol>

Given that $\mathbf{a}$ and $\mathbf{b}$ are non-zero vectors such that $\mathbf{a}=(\mathbf{a}\cdot\mathbf{b})\mathbf{b}$. State the relationship between $\mathbf{a}$ and $\mathbf{b}$. Find $|\mathbf{b}|$.

With reference to the origin $O$, the points $M$ and $N$ are such that $\overrightarrow{OM}=\alpha\mathbf{i}+4\mathbf{j}+\beta\mathbf{k}$ and $\overrightarrow{ON}=4\mathbf{i}+\alpha\mathbf{j}+\beta\mathbf{k}$, where $\alpha$ and $\beta$ are constants.

Parallel Vectors

Finding Points on a Plane

Perpendicular Vectors Using the Scalar Product

2025 NYJC Promo Q7

<p>Two vectors $\mathbf{a}$ and $\mathbf{b}$ are such that&nbsp;$\vert{}\mathbf{a} \times \mathbf{b}\vert{} = 3$. It is given that&nbsp;$\mathbf{a}$ is a unit vector and $\mathbf{b} \cdot \mathbf{b} = 9$.</p>
<p>Show that&nbsp;$\mathbf{a}$ and $\mathbf{b}$ are perpendicular.<span class="inline-marks">[3]</span></p>

Two vectors $\mathbf{a}$ and $\mathbf{b}$ are such that $\vert{}\mathbf{a} \times \mathbf{b}\vert{} = 3$. It is given that $\mathbf{a}$ is a unit vector and $\mathbf{b} \cdot \mathbf{b} = 9$.

Show that $\mathbf{a}$ and $\mathbf{b}$ are perpendicular.

N2024 P1 Q2

<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/1783577066686-nwkpkpnl7cj.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODM1NzcwNjY2ODYtbndrcGtwbmw3Y2oucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDc5MiwiZXhwIjoxNzkzNTA1MDkyfQ.Qy7roS2JdzvrnIIa0YnGRv8tkeLYScxt_m3-8hex3jk" style="max-width: min(180px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The origin $O$ and the points $A$, $B$ and $C$ lie in the same plane, where&nbsp;$\overrightarrow{OA} = \mathbf{a}$, $\overrightarrow{OB} = \mathbf{b}$ and $\overrightarrow{OC} = \mathbf{c}$. It is given that&nbsp;$\angle AOC = \angle BOC = \theta$, where&nbsp;$\theta$ is an acute angle.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;$\mathbf{c} \cdot \mathbf{\hat{a}} = \mathbf{c} \cdot \mathbf{\hat{b}}$ where&nbsp;$\mathbf{\hat{a}}$ and $\mathbf{\hat{b}}$ are unit vectors in the directions of vectors $\mathbf{a}$ and $\mathbf{b}$ respectively.<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">If&nbsp;$\mathbf{c}$ can be written as&nbsp;$m\mathbf{\hat{a}} + n\mathbf{\hat{b}}$, where&nbsp;$m$ and $n$ are constants, use the result from <strong>(a)</strong> to show that&nbsp;$m = n$.<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">Write down the equation of the line passing through the points $A$ and $B$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Given that&nbsp;$|\mathbf{a}| = 3$, $|\mathbf{b}| = 2$ and $m = n$, show that the position vector of the point of intersection of the line passing through $A$ and $B$ and the line passing through $O$ and $C$ is $t\left(\mathbf{\hat{a}} + \mathbf{\hat{b}}\right)$, where&nbsp;$t$ is a constant to be determined.<span class="inline-marks">[3]</span></div></li></ol>

The origin $O$ and the points $A$, $B$ and $C$ lie in the same plane, where $\overrightarrow{OA} = \mathbf{a}$, $\overrightarrow{OB} = \mathbf{b}$ and $\overrightarrow{OC} = \mathbf{c}$. It is given that $\angle AOC = \angle BOC = \theta$, where $\theta$ is an acute angle.

Intersection of Two Lines

2024 ACJC P2 Q4

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

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

Distance from a Point to a Plane

Equations of Planes

2024 ACJC P2 Q2

<p>Referred to the origin $O$, the points $A$, $B$ and $C$ have position vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$. $C$ lies on $AB$ such that&nbsp;$AC : AB = 1 : 3$.</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">Find $\mathbf{c}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-ii"><div class="list-item-paragraph">Show that the area of triangle $OAC$ is given by $k|\mathbf{a} \times \mathbf{b}|$, where&nbsp;$k$ is a constant to be determined.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is now given that&nbsp;$\mathbf{a}$ and $\mathbf{b}$ are unit vectors.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-iii"><div class="list-item-paragraph">Using the result in part <strong>(i)</strong>, find the value of $(2\mathbf{a} - \mathbf{b}) \cdot \mathbf{c}$.<span class="inline-marks">[3]</span></div></li></ol>

Referred to the origin $O$, the points $A$, $B$ and $C$ have position vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$. $C$ lies on $AB$ such that $AC : AB = 1 : 3$.

It is now given that $\mathbf{a}$ and $\mathbf{b}$ are unit vectors.

Ratio Theorem and Section Formula

2024 MI PRU1 Q4

<p>$\mathbf{a}$ and $\mathbf{b}$ are vectors such that&nbsp;$|\mathbf{a}| = \sqrt{2}$ and $|\mathbf{b}| = \sqrt{6}$. The angle between $\mathbf{a}$ and $\mathbf{b}$ is $\frac{5\pi}{6}$.</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Find the exact length of projection of $\mathbf{a}$ onto $\mathbf{b}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">By considering $(3\mathbf{a} + 2\mathbf{b}) \cdot (3\mathbf{a} + 2\mathbf{b})$, find the exact value of $|3\mathbf{a} + 2\mathbf{b}|$.<span class="inline-marks">[4]</span></div></li></ol>

$\mathbf{a}$ and $\mathbf{b}$ are vectors such that $|\mathbf{a}| = \sqrt{2}$ and $|\mathbf{b}| = \sqrt{6}$. The angle between $\mathbf{a}$ and $\mathbf{b}$ is $\frac{5\pi}{6}$.

2024 EJC Promo Q5

<p>Referred to the origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively such that&nbsp;$|\mathbf{a}|=5$ and $|\mathbf{b}|=3$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">It is given that&nbsp;$|2\mathbf{b} - \mathbf{a}| = 7$. Using a suitable scalar product, show that&nbsp;$\mathbf{a} \cdot \mathbf{b} = 3$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Deduce the length of projection of $\overrightarrow{OA}$ onto $\overrightarrow{OB}$.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>The point $P$ lies on $AB$ such that&nbsp;$AP:PB = 1:k$, where&nbsp;$k$ is a positive constant.</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Write down the position vector of $P$ in terms of $k$, $\mathbf{a}$ and $\mathbf{b}$ and show that the area of triangle $OAP$ is $\frac{3\sqrt{6}}{k+1}$.</span><span class="inline-marks">[5]</span></div></li></ol>

Referred to the origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively such that $|\mathbf{a}|=5$ and $|\mathbf{b}|=3$.

It is given that $|2\mathbf{b} - \mathbf{a}| = 7$. Using a suitable scalar product, show that $\mathbf{a} \cdot \mathbf{b} = 3$.

Deduce the length of projection of $\overrightarrow{OA}$ onto $\overrightarrow{OB}$.

The point $P$ lies on $AB$ such that $AP:PB = 1:k$, where $k$ is a positive constant.

Write down the position vector of $P$ in terms of $k$, $\mathbf{a}$ and $\mathbf{b}$ and show that the area of triangle $OAP$ is $\frac{3\sqrt{6}}{k+1}$.

<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>Relative to the origin $O$, the points $A$ and $C$ have position vectors $\mathbf{a}$ and $2\mathbf{b}$ such that</p>
<p style="text-align: center">\[\mathbf{a}=5p\mathbf{i}-2p\mathbf{j}+4p\mathbf{k} \text{ and}\hspace{0.5em} \mathbf{b}=\mathbf{i}-2\mathbf{j}+2\mathbf{k},\]</p>
<p>where&nbsp;$p$ is a positive constant. It is given that&nbsp;$|\mathbf{a}|=2|\mathbf{b}|$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the exact value of $p$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Evaluate $(\mathbf{a}+2\mathbf{b}) \cdot (\mathbf{a}-2\mathbf{b})$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Evaluate $\mathbf{a} \times \mathbf{b}$ and hence find the area of triangle $OAC$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Use a geometrical reason to explain why $|\mathbf{a} \times \mathbf{b}| = \frac{1}{4}(|\mathbf{a}+2\mathbf{b}|)(|\mathbf{a}-2\mathbf{b}|)$.</span><span class="inline-marks">[2]</span></div></li></ol>

Relative to the origin $O$, the points $A$ and $C$ have position vectors $\mathbf{a}$ and $2\mathbf{b}$ such that

\[\mathbf{a}=5p\mathbf{i}-2p\mathbf{j}+4p\mathbf{k} \text{ and } \mathbf{b}=\mathbf{i}-2\mathbf{j}+2\mathbf{k},\]

where $p$ is a positive constant. It is given that $|\mathbf{a}|=2|\mathbf{b}|$.

Evaluate $(\mathbf{a}+2\mathbf{b}) \cdot (\mathbf{a}-2\mathbf{b})$.

Evaluate $\mathbf{a} \times \mathbf{b}$ and hence find the area of triangle $OAC$.

Use a geometrical reason to explain why $|\mathbf{a} \times \mathbf{b}| = \frac{1}{4}(|\mathbf{a}+2\mathbf{b}|)(|\mathbf{a}-2\mathbf{b}|)$.

Vectors in a Parallelogram

2025 RI P1 Q3

<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">A triangle $ABC$ is such that&nbsp;$AB^2 + BC^2 = AC^2$.<br>By considering $\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}$ and using the fact that&nbsp;$|\mathbf{v}|^2 = \mathbf{v} \cdot \mathbf{v}$ for any vector $\mathbf{v}$, prove that&nbsp;$\angle ABC$ is a right-angle.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">In a triangle $ABC$, the point $D$ divides $AC$ in the ratio $\lambda : 1 - \lambda$, where&nbsp;$0 &lt; \lambda &lt; 1$. Let the position vectors of $A$, $B$, $C$ and $D$ be denoted by $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ and $\mathbf{d}$ respectively. Show that the area of triangle $ABD$ is given by\[k|\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a}|\]where&nbsp;$k$ is to be determined in terms of $\lambda$.<span class="inline-marks">[4]</span></div></li></ol>

2025 NYJC P1 Q3

<p>With respect to an origin $O$, points $P$ and $Q$ have variable position vectors $\mathbf{p}$ and $\mathbf{q}$ respectively, given by $\mathbf{p}=(\cos t)\mathbf{i}+(\sin t)\mathbf{j}-\mathbf{k}$, $\mathbf{q}=(\cos 2t)\mathbf{i}-(\sin 2t)\mathbf{j}+\alpha \mathbf{k}$, where&nbsp;$\alpha$ is a real parameter and $0&lt;t&lt;\pi$.</p>
<ol class="i-list"><li data-list-item-id="item-i-4" data-list-item-path="item-i-4"><div class="list-item-paragraph">By considering the cross product or otherwise, find the exact values of $t$ and $\alpha$ if&nbsp;$O$, $P$ and $Q$ are collinear.<span class="inline-marks">[5]</span></div></li></ol>
<p>It is given that&nbsp;$\alpha=0.5$.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-ii-4" data-list-item-path="item-ii-4"><div class="list-item-paragraph">Show that&nbsp;$\mathbf{p}\cdot \mathbf{q}=\cos 3t-0.5$. Find the greatest length of projection of $\mathbf{q}$ onto $\mathbf{p}$.<span class="inline-marks">[4]</span></div></li></ol>

With respect to an origin $O$, points $P$ and $Q$ have variable position vectors $\mathbf{p}$ and $\mathbf{q}$ respectively, given by $\mathbf{p}=(\cos t)\mathbf{i}+(\sin t)\mathbf{j}-\mathbf{k}$, $\mathbf{q}=(\cos 2t)\mathbf{i}-(\sin 2t)\mathbf{j}+\alpha \mathbf{k}$, where $\alpha$ is a real parameter and $0<t<\pi$.

By considering the cross product or otherwise, find the exact values of $t$ and $\alpha$ if $O$, $P$ and $Q$ are collinear.

Show that $\mathbf{p}\cdot \mathbf{q}=\cos 3t-0.5$. Find the greatest length of projection of $\mathbf{q}$ onto $\mathbf{p}$.

Collinearity Using Vectors

2025 NJC P2 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/pasted/1768703533158-y2y9mw7e2d.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg3MDM1MzMxNTgteTJ5OW13N2UyZC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0NzkyLCJleHAiOjE3OTM1MDUwOTJ9.kfIkcWCqx0qnVzR4ODc6XP1ucWXTZUfSBQe1aShRqho" 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>The points $A$ and $B$ have position vectors $4\mathbf{i}+3\mathbf{k}$ and $8\mathbf{i}-6\mathbf{k}$ respectively. The point $C$ on the line segment $AB$ is such that&nbsp;$(1-k)AC=kBC$. By using the scalar product, find the value of $k$ such that&nbsp;$OC$ bisects the angle $AOB$.</p>

The points $A$ and $B$ have position vectors $4\mathbf{i}+3\mathbf{k}$ and $8\mathbf{i}-6\mathbf{k}$ respectively. The point $C$ on the line segment $AB$ is such that $(1-k)AC=kBC$. By using the scalar product, find the value of $k$ such that $OC$ bisects the angle $AOB$.

<p>The points $O$, $A$ and $B$ are on a plane such that relative to the point $O$, the points $A$ and $B$ have non-parallel position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. The point $C$ with position vector $\mathbf{c}$ is on the plane $AOB$ such that&nbsp;$OC$ bisects the angle $AOB$.</p>
<p>Show that&nbsp;$\left(\frac{\mathbf{a}}{|\mathbf{a}|}-\frac{\mathbf{b}}{|\mathbf{b}|}\right)\cdot\mathbf{c}=0$.</p>

The points $O$, $A$ and $B$ are on a plane such that relative to the point $O$, the points $A$ and $B$ have non-parallel position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. The point $C$ with position vector $\mathbf{c}$ is on the plane $AOB$ such that $OC$ bisects the angle $AOB$.

Show that $\left(\frac{\mathbf{a}}{|\mathbf{a}|}-\frac{\mathbf{b}}{|\mathbf{b}|}\right)\cdot\mathbf{c}=0$.

<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>

<ol class="abc-list"><li><div class="list-item-paragraph">Let&nbsp;$\mathbf{u}$ and $\mathbf{v}$ be non-zero vectors such that&nbsp;$\mathbf{v}$ is a unit vector. Show that&nbsp;$|\mathbf{u}\cdot\mathbf{v}|\le|\mathbf{u}|$. When does equality occur? Explain your reasoning.<br><span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Show that if&nbsp;$\mathbf{x}$ and $\mathbf{y}$ are non-zero parallel vectors, the $\mathbf{x}\times\mathbf{y}=\mathbf{0}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Referred to the origin $O$, points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively.<br>Let&nbsp;$C$ be a variable point with position vector $\lambda(\mathbf{a}-\mathbf{b})$.<br>Show that the area of $\triangle ABC$ is independent of $\lambda$.<span class="inline-marks">[2]</span></div></li></ol>

Practice 1 – Basic Results Q11

<p>The angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ is $60^\circ$ and $|\mathbf{a}|=2$, $|\mathbf{b}|=1$. Find the value of $\mathbf{a}\cdot\mathbf{b}$.</p>
<p>By simplifying $(2\mathbf{a}-\mathbf{b})\cdot(2\mathbf{a}-\mathbf{b})$, show that&nbsp;$|2\mathbf{a}-\mathbf{b}|=\sqrt{13}$.</p>

The angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ is $60^\circ$ and $|\mathbf{a}|=2$, $|\mathbf{b}|=1$. Find the value of $\mathbf{a}\cdot\mathbf{b}$.

By simplifying $(2\mathbf{a}-\mathbf{b})\cdot(2\mathbf{a}-\mathbf{b})$, show that $|2\mathbf{a}-\mathbf{b}|=\sqrt{13}$.

<p>Two unit vectors $\mathbf{a}$ and $\mathbf{b}$ are such that the angle between $\mathbf{a}$ and $\mathbf{b}$ is $60^\circ$.</p>
<p>Given that&nbsp;$\mathbf{x}=2\mathbf{a}-\mathbf{b}$, $\mathbf{y}=-\mathbf{a}+3\mathbf{b}$  and $\mathbf{z}=3\mathbf{a}+\mathbf{b}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;$\mathbf{x}$ is perpendicular to $\mathbf{z}+3\mathbf{y}$.</div></li><li><div class="list-item-paragraph">Find the magnitude of $\mathbf{x}+2\mathbf{y}+2\mathbf{z}$.</div></li></ol>

Two unit vectors $\mathbf{a}$ and $\mathbf{b}$ are such that the angle between $\mathbf{a}$ and $\mathbf{b}$ is $60^\circ$.

Given that $\mathbf{x}=2\mathbf{a}-\mathbf{b}$, $\mathbf{y}=-\mathbf{a}+3\mathbf{b}$  and $\mathbf{z}=3\mathbf{a}+\mathbf{b}$.

Math Question Bank