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<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

Scalar (Dot) Product

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Vectors

<p>The diagram shows a unit circle with centre $O$ and diameter $AB$. The point $C$ lies on the circumference of the circle. The position vectors of $A$, $B$ and $C$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively.</p>
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<ol class="abc-list"><li data-list-item-id="h2-2025-cjc-promo-q6-cjc-q6-item-block-2-item-0" data-list-item-path="h2-2025-cjc-promo-q6-cjc-q6-item-block-2-item-0"><div class="list-item-paragraph">By considering an appropriate scalar product, show that&nbsp;$AC$ and $BC$ are perpendicular.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="h2-2025-cjc-promo-q6-cjc-q6-item-block-2-item-1" data-list-item-path="h2-2025-cjc-promo-q6-cjc-q6-item-block-2-item-1"><div class="list-item-paragraph">Give a geometrical interpretation of $|\mathbf{a}\cdot(\mathbf{c}-\mathbf{a})|$.<span class="inline-marks">[1]</span></div></li></ol>
<p>The point $D$ is such that&nbsp;$ABCD$ is a parallelogram. The point $M$ is the mid-point of $AD$ and the point $R$ lies on $OA$ produced such that&nbsp;$OA:AR=1:2$.</p>
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<ol class="abc-list" start="3"><li data-list-item-id="h2-2025-cjc-promo-q6-cjc-q6-item-block-5-item-0" data-list-item-path="h2-2025-cjc-promo-q6-cjc-q6-item-block-5-item-0"><div class="list-item-paragraph">By considering $\overrightarrow{CM}\times\overrightarrow{CR}$ or otherwise, show that the points $C$, $M$ and $R$ are collinear.<span class="inline-marks">[3]</span></div></li></ol>

The diagram shows a unit circle with centre $O$ and diameter $AB$. The point $C$ lies on the circumference of the circle. The position vectors of $A$, $B$ and $C$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively.

By considering an appropriate scalar product, show that $AC$ and $BC$ are perpendicular.

Give a geometrical interpretation of $|\mathbf{a}\cdot(\mathbf{c}-\mathbf{a})|$.

The point $D$ is such that $ABCD$ is a parallelogram. The point $M$ is the mid-point of $AD$ and the point $R$ lies on $OA$ produced such that $OA:AR=1:2$.

By considering $\overrightarrow{CM}\times\overrightarrow{CR}$ or otherwise, show that the points $C$, $M$ and $R$ are collinear.

Collinearity Using Vectors

Vectors in a Parallelogram

Perpendicular Vectors Using the Scalar Product

Ratio Theorem and Section Formula

2025 CJC Promo Q6

<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>

Geometrical Interpretation of Vector Equations

2024 NJC Promo Q7

<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 $\frac{5\pi}{6}$.

Magnitude of a Vector

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>Referred to the origin $O$, points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. Point $C$ lies on $OA$, such that&nbsp;$OC:CA=2:1$. Point $D$ lies on $OB$, such that&nbsp;$OD:DB=\lambda:\mu$. It is given that the area of triangle $ABD$ is half the area of triangle $ABC$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show the area of triangle $ABD$ is given by $\frac{\mu}{2(\lambda+\mu)} |\mathbf{a} \times \mathbf{b}|$. Hence find the ratio $\lambda:\mu$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">The point $E$ has position vector $\frac{1}{4}\mathbf{a}+\frac{5}{8}\mathbf{b}$. Show that&nbsp;$A$, $E$ and $D$ are collinear.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>It is further given that the angle $AOB$ is $\frac{\pi}{4}$ and $O$ lies on the perpendicular bisector of the line segment $AB$.</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the length of projection of $\mathbf{a}$ on $\mathbf{b}$, giving your answer in terms of $|\mathbf{b}|$. Hence find the position vector of the point $F$, the foot of perpendicular from $A$ to $OB$.</span><span class="inline-marks">[3]</span></div></li></ol>

Referred to the origin $O$, points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. Point $C$ lies on $OA$, such that 
$OC:CA=2:1$. Point $D$ lies on $OB$, such that $OD:DB=\lambda:\mu$. It is given that the area of triangle $ABD$ is half the area of triangle $ABC$.

The point $E$ has position vector $\frac{1}{4}\mathbf{a}+\frac{5}{8}\mathbf{b}$. Show that $A$, $E$ and $D$ are collinear.

It is further given that the angle $AOB$ is $\frac{\pi}{4}$ and $O$ lies on the perpendicular bisector of the line segment $AB$.

Find the length of projection of $\mathbf{a}$ on $\mathbf{b}$, giving your answer in terms of $|\mathbf{b}|$. Hence find the position vector of the point $F$, the foot of perpendicular from $A$ to $OB$.

Perpendicular Projection and Reflection in a Line

2025 VJC 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 $\mathbf{p}$.

2025 NJC P2 Q3

<p>Referred to the origin $O$, the 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 vectors. The point $C$ lies on $OA$ such that&nbsp;$OC : CA = 1 : 2$. The point $D$ lies on $OB$ such that&nbsp;$OB : DB = 3 : 1$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the position vectors $\overrightarrow{OC}$ and $\overrightarrow{OD}$, giving your answers in terms of $\mathbf{a}$ and $\mathbf{b}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that the point $E$ where the lines $AD$ and $BC$ meet has position vector \[\frac{1}{7}\mathbf{a} + \frac{4}{7}\mathbf{b}\]</span><span class="inline-marks">[5]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that the area of triangle $OAE$ can be written as&nbsp;$k|\mathbf{a} \times \mathbf{b}|$, where&nbsp;$k$ is a constant to be found.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>It is further given that&nbsp;$\mathbf{b}$ is a unit vector.</p>
<ol class="i-list" start="4"><li><div class="list-item-content"><span class="list-item-text">Give the geometrical meaning of $|\mathbf{a} \cdot \mathbf{b}|$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">If&nbsp;$|\mathbf{a} \cdot \mathbf{b}| = \frac{1}{2}|\mathbf{a}|$, find the perpendicular distance of point $A$ to $OB$, leaving your answer in terms of $|\mathbf{a}|$.</span><span class="inline-marks">[3]</span></div></li></ol>

Referred to the origin $O$, the 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 vectors. The point $C$ lies on $OA$ such that $OC : CA = 1 : 2$. The point $D$ lies on $OB$ such that $OB : DB = 3 : 1$.

Show that the point $E$ where the lines $AD$ and $BC$ meet has position vector \[\frac{1}{7}\mathbf{a} + \frac{4}{7}\mathbf{b}\]

Show that the area of triangle $OAE$ can be written as $k|\mathbf{a} \times \mathbf{b}|$, where $k$ is a constant to be found.

It is further given that $\mathbf{b}$ is a unit vector.

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

If $|\mathbf{a} \cdot \mathbf{b}| = \frac{1}{2}|\mathbf{a}|$, find the perpendicular distance of point $A$ to $OB$, leaving your answer in terms of $|\mathbf{a}|$.

Intersection of Two Lines

Distance from a Point to a Line

2025 MI P1 Q11

<p>Relative to the origin $O$, point $A$ has position vector given by $\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$. The line $l_1$ passes through the point $A$ and is parallel to $\begin{pmatrix} t \\ 3 - t^2 \\ 1 \end{pmatrix}$. The plane $p$ has equation $\mathbf{r} = \lambda \begin{pmatrix} 0 \\ 5 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 5 \\ 0 \\ t \end{pmatrix}, \lambda, \mu \in \mathbb{R}$ where&nbsp;$t$ is a real constant. It is known that&nbsp;$l_1$ and $p$ are parallel and $l_1$ is not on $p$.</p>
<ol class="abc-list"><li data-list-item-id="item-1768711344529-8nihmues6" data-list-item-path="item-1768711344529-8nihmues6"><div class="list-item-paragraph">Show that&nbsp;$t = -1$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1768711345279-s547r63ey" data-list-item-path="item-1768711345279-s547r63ey"><div class="list-item-paragraph">The line $l_2$ passes through the origin and point $A$. Find the acute angle between $l_1$ and $l_2$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1768711345923-79fmat0do" data-list-item-path="item-1768711345923-79fmat0do"><div class="list-item-paragraph">The vector $\mathbf{n}$ is a unit vector normal to $p$. State the geometrical meaning of $|\mathbf{a} \cdot \mathbf{n}|$ and find the exact value of $|\mathbf{a} \cdot \mathbf{n}|$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1768711346929-qm8zctapw" data-list-item-path="item-1768711346929-qm8zctapw"><div class="list-item-paragraph">Find a vector equation of the line of reflection of $l_2$ in $p$.<span class="inline-marks">[4]</span></div></li></ol>

Relative to the origin $O$, point $A$ has position vector given by $\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$. The line $l_1$ passes through the point $A$ and is parallel to $\begin{pmatrix} t \\ 3 - t^2 \\ 1 \end{pmatrix}$. The plane $p$ has equation $\mathbf{r} = \lambda \begin{pmatrix} 0 \\ 5 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 5 \\ 0 \\ t \end{pmatrix}, \lambda, \mu \in \mathbb{R}$ where $t$ is a real constant. It is known that $l_1$ and $p$ are parallel and $l_1$ is not on $p$.

Perpendicular Projection and Reflection in a Plane

Distance from a Point to a Plane

Angles Between Two Lines

Relationships Between a Line and a Plane

2025 EJC P2 Q4

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg3MDM1MzMxNTgteTJ5OW13N2UyZC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA4NjM0LCJleHAiOjE3OTM1MDg5MzR9.517xvID3VadIrqmT9aVnsemCfq51CEtNV7LO_8C6zqk" 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>Relative to the origin $O$, the position vectors of points $A$, $B$ and $C$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively. It is given that&nbsp;$\mathbf{a}$ is a unit vector and $\mathbf{b}$ is perpendicular to $\mathbf{b} - \mathbf{a}$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$0 &lt; |\mathbf{b}| &lt; 1$.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>It is given further that&nbsp;$|\mathbf{b}| = \frac{1}{2}$ and $C$ lies on $AB$ such that&nbsp;$AC : CB = 2 : 1$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find the value of $|\mathbf{a} \cdot \mathbf{c}|$ and state the geometrical interpretation of $|\mathbf{a} \cdot \mathbf{c}|$.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the value of $\frac{|\mathbf{a} \times \mathbf{b}|}{|\mathbf{a} \times \mathbf{c}|}$.</span><span class="inline-marks">[2]</span></div></li></ol>

Relative to the origin $O$, the position vectors of points $A$, $B$ and $C$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively. It is given that $\mathbf{a}$ is a unit vector and $\mathbf{b}$ is perpendicular to $\mathbf{b} - \mathbf{a}$.

It is given further that $|\mathbf{b}| = \frac{1}{2}$ and $C$ lies on $AB$ such that $AC : CB = 2 : 1$.

Find the value of $|\mathbf{a} \cdot \mathbf{c}|$ and state the geometrical interpretation of $|\mathbf{a} \cdot \mathbf{c}|$.

Find the value of $\frac{|\mathbf{a} \times \mathbf{b}|}{|\mathbf{a} \times \mathbf{c}|}$.

2025 CJC P1 Q6

<p>The equation of line $l$ is given by $\mathrm{r}=\left( \begin{matrix}2\\1\\0 \\\end{matrix}\right)+\lambda \left( \begin{matrix}1\\-2\\1\\\end{matrix} \right)$. The coordinates of points $A$ and $B$ are $\left( 1,3,-2 \right)$ and$(1,0,-2)$. Find the length of projection of $\overrightarrow{AB}$ on $l$.</p>

The equation of line $l$ is given by $\mathrm{r}=\left( \begin{matrix}2\\1\\0 \\\end{matrix}\right)+\lambda \left( \begin{matrix}1\\-2\\1\\\end{matrix} \right)$. The coordinates of points $A$ and $B$ are $\left( 1,3,-2 \right)$ and$(1,0,-2)$. Find the length of projection of $\overrightarrow{AB}$ on $l$.

Unit 13.1 Worked Example 5

<p>The position vectors of points $P$, $Q$ and $R$ relative to an origin $O$ are $-4\mathbf{i}-4\mathbf{j}$, $2\mathbf{i}+6\mathbf{j}$ and $5\mathbf{i}-2\mathbf{j}$ respectively. Find</p>
<ol class="i-list"><li><div class="list-item-paragraph">the exact perpendicular distance of the point $P$ from the line segment $QR$.</div></li><li><div class="list-item-paragraph">the exact projection length of $\overrightarrow{QP}$ on $\overrightarrow{QR}$.</div></li></ol>

The position vectors of points $P$, $Q$ and $R$ relative to an origin $O$ are $-4\mathbf{i}-4\mathbf{j}$, $2\mathbf{i}+6\mathbf{j}$ and $5\mathbf{i}-2\mathbf{j}$ respectively. Find

<p>The coordinates of points $A$, $B$ and $C$ are $(-2,5,0)$, $(0,-2,6)$ and $(1,0,-2)$. Find the length of projection of $\overrightarrow{AB}$ on $\overrightarrow{AC}$. Hence, find the shortest distance of point $B$ to line $AC$.</p>

The coordinates of points $A$, $B$ and $C$ are $(-2,5,0)$, $(0,-2,6)$ and $(1,0,-2)$. Find the length of projection of $\overrightarrow{AB}$ on $\overrightarrow{AC}$. Hence, find the shortest distance of point $B$ to line $AC$.

<p>The equation of line $l$ is given by $\mathrm{r}=\left( \begin{matrix}2\\1\\0 \\\end{matrix}\right)+\lambda \left( \begin{matrix}1\\-2\\1\\\end{matrix} \right)$. The coordinates of points $A$ and $B$ are $\left( 1,3,-2 \right)$ and $(1,0,-2)$. Find the length of projection of $\overrightarrow{AB}$ on $l$.</p>

The equation of line $l$ is given by $\mathrm{r}=\left( \begin{matrix}2\\1\\0 \\\end{matrix}\right)+\lambda \left( \begin{matrix}1\\-2\\1\\\end{matrix} \right)$. The coordinates of points $A$ and $B$ are $\left( 1,3,-2 \right)$ and $(1,0,-2)$. Find the length of projection of $\overrightarrow{AB}$ on $l$.

<p>Referred to the origin $O$, 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 vectors. Point $C$ is the midpoint of $OA$ and point $D$ lies on $BC$ such that&nbsp;$BD:DC=2:3$.</p>
<ol class="i-list"><li data-list-item-id="item-1761382899263-brvks20tx" data-list-item-path="item-1761382899263-brvks20tx"><div class="list-item-paragraph">Find the position vectors $\overrightarrow{OC}$ and $\overrightarrow{OD}$, giving your answers in terms of $\mathbf{a}$ and $\mathbf{b}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1761382900418-mevh1bxll" data-list-item-path="item-1761382900418-mevh1bxll"><div class="list-item-paragraph">Find the area of triangle $OCD$ in the form of $k\left| \mathbf{a}\times \mathbf{b} \right|$, where&nbsp;$k$ is a constant to be found.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1761382901431-gjpcpaoik" data-list-item-path="item-1761382901431-gjpcpaoik"><div class="list-item-paragraph">Given that&nbsp;$\left| \mathbf{a} \right|=2$, $\left| \mathbf{b} \right|=\sqrt{2}$ and $\angle AOB=\frac{\pi }{4}$ radian, using scalar product, show that&nbsp;$CD$ is perpendicular to $OA$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1761382903185-eppje5jgm" data-list-item-path="item-1761382903185-eppje5jgm"><div class="list-item-paragraph">Given instead that&nbsp;$\mathbf{a}$ is a unit vector, give a geometrical interpretation of $\left| \mathbf{a}\cdot \mathbf{b} \right|$.<span class="inline-marks">[1]</span></div></li></ol>

Referred to the origin $O$, 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 vectors. Point $C$ is the midpoint of $OA$ and point $D$ lies on $BC$ such that $BD:DC=2:3$.

2024 JPJC Promo Q11

<p>Relative to the origin $O$, 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 vectors. The point $M$ is the midpoint of $OA$ and the point $N$ lies on $OB$ such that $ON:NB=1:2$. The point $C$ with position vector $\mathbf{c}$ is the point of intersection of lines $AN$ and $BM$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $\mathbf{c}=\frac{2}{5}\mathbf{a}+\frac{1}{5}\mathbf{b}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Give the geometrical meaning of $\left| \mathbf{a}\cdot \widehat{\mathbf{c}} \right|$, where $\widehat{\mathbf{c}}$ is the unit vector in the direction of $\mathbf{c}$.</div><p style="text-align: right;">[1]</p></li></ol>
<p>The points $O$, $A$ and $C$ lies on a circle where $OA$ is the diameter of the circle. It is further given that $\mathbf{a}\cdot \mathbf{b}=\frac{1}{2}{{\left| \mathbf{a} \right|}^{2}}$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find the ratio of $\left| \mathbf{a} \right|:\left| \mathbf{c} \right|$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Hence find the area of triangle $OAC$, giving your answer in terms of $\left| \mathbf{a} \right|$.</div><p style="text-align: right;">[2]</p></li></ol>

Relative to the origin $O$, 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 vectors. The point $M$ is the midpoint of $OA$ and the point $N$ lies on $OB$ such that $ON:NB=1:2$. The point $C$ with position vector $\mathbf{c}$ is the point of intersection of lines $AN$ and $BM$.

The points $O$, $A$ and $C$ lies on a circle where $OA$ is the diameter of the circle. It is further given that $\mathbf{a}\cdot \mathbf{b}=\frac{1}{2}{{\left| \mathbf{a} \right|}^{2}}$.

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<p>The coordinates of points $A$ and $B$ are $\left( 2,-3,2 \right)$ and $(3,-4,4)$. Find the exact length of projection of $\overrightarrow{AB}$ on the plane ${{p}_{1}}:3x-2y-3z=4$.</p>

The coordinates of points $A$ and $B$ are $\left( 2,-3,2 \right)$ and $(3,-4,4)$. Find the exact length of projection of $\overrightarrow{AB}$ on the plane ${{p}_{1}}:3x-2y-3z=4$.

<p>The coordinates of points $A$ and $B$ are $\left( 3,1,-2 \right)$ and $\left( 7,4,-5 \right)$. Find the exact length of projection of $\overrightarrow{AB}$ on the plane ${{p}_{1}}:x-3y+2z=-5$.</p>

The coordinates of points $A$ and $B$ are $\left( 3,1,-2 \right)$ and $\left( 7,4,-5 \right)$. Find the exact length of projection of $\overrightarrow{AB}$ on the plane ${{p}_{1}}:x-3y+2z=-5$.

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<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: left; --cell-justify: flex-start"><h3><strong>Relationships Between</strong></h3></th><th style="text-align: left; --cell-justify: flex-start"><h3>Essential Skills Checklist</h3></th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start"><h3>A point and a line</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Shortest distance of the point to the line</p><p>☐ Foot of perpendicular</p><p>☐ Mirror image of a point in a line</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>A line segment and a line</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Length of projection of a line segment on a line</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>$2$ parallel lines</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ $2$ parallel lines</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>$2$ Intersecting Lines</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Point of intersection</p><p>☐ Angle between $2$ intersecting lines</p><p>☐ Mirror image of a line in another line</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>$2$ skew Lines</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Angle between them</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>A point and a plane</h3><h3></h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Deduce a random point on plane</p><p>☐ Shortest distance of point to plane</p><p>☐ Foot of perpendicular on the plane</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>The origin and a plane</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Equations of planes which contains the origin</p><p>☐ $x-y$, $y-z$ and $x-z$ planes</p><p>☐ Shortest distance of origin to the plane</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>Line segment and a plane</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Length of projection of line segment to the plane</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>Line and a plane</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Point of intersection</p><p>☐ Angle between the line and plane</p><p>☐ Mirror Image of line in plane</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>$2$ Intersecting Planes</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Equation of intersecting line without using GC</p><p>☐ Equation of intersecting line by using GC</p><p>☐ Angle between the planes</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"><h3>$2$ Parallel Planes</h3></td><td style="text-align: left; --cell-justify: flex-start"><p>☐ Distance between $2$ parallel planes</p><p>☐ Finding equations of planes a fixed distance away from another parallel plane</p></td></tr></tbody></table></div></div>

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