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PSLE

PSLE Foundation Mathematics

<p>In the diagram, $OABC$ is a parallelogram. $D$ is the mid-point of $BC$. It is given that&nbsp;$\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OC} = \mathbf{c}$. $E$ is the point on $OC$ such that&nbsp;$5OE = 2OC$. $F$ is the point on $OD$ such that&nbsp;$\overrightarrow{OF} = \frac{1}{2}\overrightarrow{FD}$.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="list-item-paragraph">Express, as simply as possible, in terms of $\mathbf{a}$ and/or $\mathbf{c}$,</div><ol class="i-list"><li><div class="list-item-paragraph">$\overrightarrow{EC}$,</div></li><li><div class="list-item-paragraph">$\overrightarrow{OD}$,</div></li><li><div class="list-item-paragraph">$\overrightarrow{OF}$.</div></li></ol></li><li><div class="list-item-paragraph">Show that&nbsp;$\overrightarrow{AE} = 6\overrightarrow{FE}$.</div></li><li><div class="list-item-paragraph">Given that the coordinates of $F$ are $(4,2)$ and $\overrightarrow{AE} = \begin{pmatrix} 3 \\ -12 \end{pmatrix}$.</div><ol class="i-list"><li><div class="list-item-paragraph">Calculate $|\overrightarrow{AE}|$.</div></li><li><div class="list-item-paragraph">Find the coordinates of $E$.</div></li><li><div class="list-item-paragraph">Find the value of $t$, given that the vector $\begin{pmatrix} 4.5 \\ t \end{pmatrix}$ is parallel to $\overrightarrow{AE}$.</div></li></ol></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1768037372726-zd47zjcpps.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgwMzczNzI3MjYtemQ0N3pqY3Bwcy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0Nzc4LCJleHAiOjE3OTM1MDUwNzh9.h3wB8z56LBllGZzfQ1BK5ClyLe1hI6FvXqUWmVAu-SM" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

In the diagram, $OABC$ is a parallelogram. $D$ is the mid-point of $BC$. It is given that $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OC} = \mathbf{c}$. $E$ is the point on $OC$ such that $5OE = 2OC$. $F$ is the point on $OD$ such that $\overrightarrow{OF} = \frac{1}{2}\overrightarrow{FD}$.

Collinearity Using Vectors

Parallel Vectors

Magnitude of a Vector

Position Vectors

Vectors in a Parallelogram

Ratio Theorem and Section Formula

Vector Properties in Geometric Shapes: The Parallelogram

Vectors

<p>The points $A$, $B$ and $C$ are vertices of a triangle. The point $A$ is $(-1,3)$ and the gradient of the line $AB$ is $2$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the equation of the line $AB$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">The equation of the line $BC$ is $2x+y=17$. Find the coordinates of $B$.</span></div></li><li><div class="list-item-paragraph">It is given that&nbsp;$\overrightarrow{AE} = \begin{pmatrix} 4 \\ 12 \end{pmatrix}$. Find the coordinates of $E$.</div></li><li><div class="list-item-content"><span class="list-item-text">Given also that&nbsp;$D$ is $(3,7)$, express as a column vector:</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$\overrightarrow{DE}$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\overrightarrow{DB}$.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">What two facts can be deduced about $D$, $E$ and $B$.</span></div></li></ol>

The points $A$, $B$ and $C$ are vertices of a triangle. The point $A$ is $(-1,3)$ and the gradient of the line $AB$ is $2$.

The equation of the line $BC$ is $2x+y=17$. Find the coordinates of $B$.

Given also that $D$ is $(3,7)$, express as a column vector:

What two facts can be deduced about $D$, $E$ and $B$.

Vector Subtraction

Coordinate Geometry and Vector Applications

<p>The coordinates of point $A$ and point $B$ are $(5,3)$ and $(2.5,-1)$ respectively.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$\overrightarrow{OC} = \overrightarrow{OA} - 3\overrightarrow{OB}$, find the column vector $\overrightarrow{OC}$.</span></div></li><li><div class="list-item-paragraph">$\overrightarrow{ST} = \begin{pmatrix} a \\ b \end{pmatrix}$, $|\overrightarrow{ST}| = 13$ units and $\overrightarrow{ST}$ is parallel to $\overrightarrow{OC}$. Find the value of $a$ and of $b$ if&nbsp;$a &gt; 0$.</div></li></ol>

The coordinates of point $A$ and point $B$ are $(5,3)$ and $(2.5,-1)$ respectively.

Given that $\overrightarrow{OC} = \overrightarrow{OA} - 3\overrightarrow{OB}$, find the column vector $\overrightarrow{OC}$.

Vector Algebra

Resultant Vector Calculations in Cartesian Form

<p>It is given that&nbsp;$A$ is the point $(3,4)$ and $B$ is the point $(-5,1)$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find $|\overrightarrow{AB}|$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">If&nbsp;$\overrightarrow{CD} = \frac{1}{2}\overrightarrow{AB}$ and $D$ is the point $(7,2)$, find the coordinates of the point $C$.</span></div></li><li><div class="list-item-paragraph">If&nbsp;$\overrightarrow{XY} = \begin{pmatrix} -2 \\ k \end{pmatrix}$ and $\overrightarrow{XY}$ is parallel to $\overrightarrow{AB}$, find the value of $k$.</div></li></ol>

It is given that $A$ is the point $(3,4)$ and $B$ is the point $(-5,1)$.

If $\overrightarrow{CD} = \frac{1}{2}\overrightarrow{AB}$ and $D$ is the point $(7,2)$, find the coordinates of the point $C$.

Algebraic Manipulation of Planar Vectors

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Given the coordinates $A(-1,3)$ and $B(0,6)$, express $\overrightarrow{AB}$ as a column vector.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, evaluate $\left| \overrightarrow{AB} \right|$.</span></div></li></ol>

Given the coordinates $A(-1,3)$ and $B(0,6)$, express $\overrightarrow{AB}$ as a column vector.

Hence, evaluate $\left| \overrightarrow{AB} \right|$.

Operations on Vectors: Addition and Subtraction

<p>Three points $O$, $P$ and $Q$ are such that $\vec{OP}=-2\mathbf{p}$ and $\vec{OQ}=3\mathbf{q}$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Express $\vec{PQ}$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Another point $R$ is such that $\vec{QR}=2\mathbf{q}-3\mathbf{p}$, find $\vec{OR}$ in terms of $\mathbf{p}$ and $\mathbf{q}$.</span></div></li></ol>

Three points $O$, $P$ and $Q$ are such that $\vec{OP}=-2\mathbf{p}$ and $\vec{OQ}=3\mathbf{q}$.

Another point $R$ is such that $\vec{QR}=2\mathbf{q}-3\mathbf{p}$, find $\vec{OR}$ in terms of $\mathbf{p}$ and $\mathbf{q}$.

Vector Addition

Calculating Directed Line Segments from Position Vectors

<p>A point $A(-5, 3)$ is translated by the translation vector $\overrightarrow{AB} = \begin{pmatrix} 6 \\ -7 \end{pmatrix}$ to the point $B$. Find the coordinates of $B$.</p>

A point $A(-5, 3)$ is translated by the translation vector $\overrightarrow{AB} = \begin{pmatrix} 6 \\ -7 \end{pmatrix}$ to the point $B$. Find the coordinates of $B$.

Defining Position Vectors from Cartesian Coordinates

<p>Three points $O$, $P$ and $Q$ are such that $\vec{OP}=3\mathbf{p}$ and $\vec{OQ}=6\mathbf{q}$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Express $\vec{PQ}$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Another point $R$ is such that $\vec{QR}=2\mathbf{q}-\mathbf{p}$, find $\vec{OR}$ in terms of $\mathbf{p}$ and $\mathbf{q}$.</span></div></li></ol>

Three points $O$, $P$ and $Q$ are such that $\vec{OP}=3\mathbf{p}$ and $\vec{OQ}=6\mathbf{q}$.

Another point $R$ is such that $\vec{QR}=2\mathbf{q}-\mathbf{p}$, find $\vec{OR}$ in terms of $\mathbf{p}$ and $\mathbf{q}$.

Introduction to Origin-Based Position Vectors

<p>Write down the position vector of $P(3, -5)$ and of $Q(5, 1)$. Then express $\overrightarrow{PQ}$ as a column vector.</p>

Write down the position vector of $P(3, -5)$ and of $Q(5, 1)$. Then express $\overrightarrow{PQ}$ as a column vector.

Introduction to Position Vectors in a Cartesian Plane

<p>The $1$ unit by $1$ unit square grid below shows the positions of point $A$, $B$ and $P$.</p>
<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/1768014954667-x9hxq2rvr68.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgwMTQ5NTQ2NjcteDloeHEycnZyNjgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDc3OCwiZXhwIjoxNzkzNTA1MDc4fQ.g6OxbSLSoJ3I20CDAlr-nikDlP-rXNnI4f7Wia07eNY" style="max-width: min(450px, 100%); height: auto; display: block;" alt="A coordinate grid showing the relative positions of points P, B, and A." class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Express $\overrightarrow{AB}$ and $\overrightarrow{PB}$ as column vectors.</span></div></li><li><div class="list-item-content"><span class="list-item-text">$Q$ is a point such that&nbsp;$ABQP$ is a parallelogram. Express $\overrightarrow{BQ}$ as a column vector.</span></div></li><li><div class="list-item-content"><span class="list-item-text">$R$ is a point such that&nbsp;$ABPR$ is a parallelogram. Express $\overrightarrow{BR}$ as a column vector.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Do the two vectors $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ have the same magnitude? Is $\overrightarrow{PQ} = \overrightarrow{PR}$? Explain.</span></div></li></ol>

The $1$ unit by $1$ unit square grid below shows the positions of point $A$, $B$ and $P$.

Express $\overrightarrow{AB}$ and $\overrightarrow{PB}$ as column vectors.

$Q$ is a point such that $ABQP$ is a parallelogram. Express $\overrightarrow{BQ}$ as a column vector.

$R$ is a point such that $ABPR$ is a parallelogram. Express $\overrightarrow{BR}$ as a column vector.

Do the two vectors $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ have the same magnitude? Is $\overrightarrow{PQ} = \overrightarrow{PR}$? Explain.

Equal Vectors

Vector Properties of Parallelograms on a Square Grid

<p>The $1$ unit by $1$ unit square grid below shows the positions of point $A$, $B$ and $P$.</p>
<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/1768014503595-lr3o66lkpb.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgwMTQ1MDM1OTUtbHIzbzY2bGtwYi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0Nzc4LCJleHAiOjE3OTM1MDUwNzh9.vBLIiT49JPzOooKLE8DTaZm4VqV6r5awX3urLoyRKQQ" style="max-width: min(450px, 100%); height: auto; display: block;" alt="A coordinate grid showing points P, A, and B." class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Express $\overrightarrow{AB}$ and $\overrightarrow{PB}$ as column vectors.</span></div></li><li><div class="list-item-content"><span class="list-item-text">$Q$ is a point such that&nbsp;$ABQP$ is a parallelogram. Express $\overrightarrow{BQ}$ as a column vector.</span></div></li><li><div class="list-item-content"><span class="list-item-text">$R$ is a point such that&nbsp;$ABPR$ is a parallelogram. Express $\overrightarrow{BR}$ as a column vector.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Do the two vectors $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ have the same magnitude? Is $\overrightarrow{PQ} = \overrightarrow{PR}$? Explain.</span></div></li></ol>

Vector Geometry within a Cartesian Plane

<p>The points $A$ and $B$ have position vectors $\left( \begin{matrix}
   -1  \\
   2  \\
   5  \\
\end{matrix} \right)$ and $\left( \begin{matrix}
   1  \\
   -2  \\
   8  \\
\end{matrix} \right)$ respectively. The point $C$ is such that&nbsp;$\overrightarrow{BC}=2\overrightarrow{AB}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the position vector of $C$.<span class="inline-marks">[2]</span></div></li></ol>
<p>The point $D$ has position vector $\left( \begin{matrix}
   1  \\
   2  \\
   d  \\
\end{matrix} \right)$ and is such that&nbsp;$\left| \overrightarrow{AD} \right|=\left| \overrightarrow{BD} \right|$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Find the value of $d$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Use a scalar product to find angle $ADB$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find exactly the position vector of the point $P$, where&nbsp;$P$ is the centre of the circle that passes through $A$,$B$ and $D$.<span class="inline-marks">[5]</span></div></li></ol>

The points $A$ and $B$ have position vectors $\left( \begin{matrix}
   -1  \\
   2  \\
   5  \\
\end{matrix} \right)$ and $\left( \begin{matrix}
   1  \\
   -2  \\
   8  \\
\end{matrix} \right)$ respectively. The point $C$ is such that $\overrightarrow{BC}=2\overrightarrow{AB}$.

The point $D$ has position vector $\left( \begin{matrix}
   1  \\
   2  \\
   d  \\
\end{matrix} \right)$ and is such that $\left| \overrightarrow{AD} \right|=\left| \overrightarrow{BD} \right|$.

Angles Between Vectors

Perpendicular Vectors Using the Scalar Product

N2023 P2 Q3

<p>Relative to the origin $O$, the position vectors of the points $A$, $B$ and $C$ are $4\mathbf{i}+\mathbf{j}+3\mathbf{k}$, $-6\mathbf{i}+3\mathbf{j}-7\mathbf{k}$ and $-2\mathbf{i}+7\mathbf{j}-6\mathbf{k}$ respectively. The plane $p$ contains the points $A$ and $C$. $p$ is also parallel to the vector $\left( \begin{matrix}
-3 \\
6 \\
2 \\
\end{matrix} \right)$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the cartesian equation of $p$ can be expressed as $2x-y+6z=25$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Find the coordinates of the foot of the perpendicular from $B$ to $p$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Hence, or otherwise, find the coordinates of the point of reflection of $B$ about $p$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">The point $M$ lies on $AB$ extended such that $AB:AM=3:5$. Find $\overrightarrow{AB}$ and determine the exact shortest distance between $M$ and $p$.</div><p style="text-align: right;">[4]</p></li></ol>

Relative to the origin $O$, the position vectors of the points $A$, $B$ and $C$ are $4\mathbf{i}+\mathbf{j}+3\mathbf{k}$, $-6\mathbf{i}+3\mathbf{j}-7\mathbf{k}$ and $-2\mathbf{i}+7\mathbf{j}-6\mathbf{k}$ respectively. The plane $p$ contains the points $A$ and $C$. $p$ is also parallel to the vector $\left( \begin{matrix}
-3 \\
6 \\
2 \\
\end{matrix} \right)$.

Perpendicular Projection and Reflection in a Plane

Distance from a Point to a Plane

Equations of Planes

2022 TMJC J1 MYE Q8

<p>The points $P$ and $Q$ have position vectors $\mathbf{a}+\mathbf{b}$ and $3\mathbf{a}-2\mathbf{b}$ respectively relative to the origin $O$.</p>
<p>Given that $OPQR$ is a parallelogram, express the vectors $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.</p>
<p>By evaluating two scalar products, show that if $OPQR$ is a square, then ${{\left| \mathbf{a} \right|}^{2}}=2{{\left| \mathbf{b} \right|}^{2}}$.</p>

The points $P$ and $Q$ have position vectors $\mathbf{a}+\mathbf{b}$ and $3\mathbf{a}-2\mathbf{b}$ respectively relative to the origin $O$.

Given that $OPQR$ is a parallelogram, express the vectors $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

By evaluating two scalar products, show that if $OPQR$ is a square, then ${{\left| \mathbf{a} \right|}^{2}}=2{{\left| \mathbf{b} \right|}^{2}}$.

Scalar (Dot) Product

Parallelogram and Square

<p>$OABC$ is a trapezium such that&nbsp;$OA$ is parallel to $CB$, and $CB$ : $OA$ = $k:1$ , where&nbsp;$k$ is a positive constant, and $k\ne 1$. Given that&nbsp;$\overrightarrow{OA}$ = $\mathbf{a}$, $\overrightarrow{OB}$ = $\mathbf{b}$, and $X$ and $Y$ are the midpoints of $OB$ and $AC$ respectively, find the following vectors in terms of $k$, $\mathbf{a}$ and $\mathbf{b}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\overrightarrow{OC},$  <span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">$\overrightarrow{OY}.$<span class="inline-marks">[2]</span></div></li></ol>
<p>Hence show that&nbsp;$XY$ is parallel to $OA$.<span class="inline-marks">[2]</span></p>
<p>It is given that&nbsp;$OB$ and $AC$ intersect at the point $D$. Find the ratio, in terms of $k$, between the area of the triangle $XYD$ and the area of the triangle $BCD$.<span class="inline-marks">[2]</span></p>

$OABC$ is a trapezium such that $OA$ is parallel to $CB$, and $CB$ : $OA$ = $k:1$ , where $k$ is a positive constant, and $k\ne 1$. Given that $\overrightarrow{OA}$ = $\mathbf{a}$, $\overrightarrow{OB}$ = $\mathbf{b}$, and $X$ and $Y$ are the midpoints of $OB$ and $AC$ respectively, find the following vectors in terms of $k$, $\mathbf{a}$ and $\mathbf{b}$.

Hence show that $XY$ is parallel to $OA$.

It is given that $OB$ and $AC$ intersect at the point $D$. Find the ratio, in terms of $k$, between the area of the triangle $XYD$ and the area of the triangle $BCD$.

Intersection of Two Lines

Areas of Triangles and Parallelograms Using Vectors

2013 RI P1 Q5

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<p>The points $A$, $B$ and $C$ lie on a circle with center $O$ and diameter $AC$. It is given that&nbsp;$\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$.</p>
<ol class="i-list"><li data-list-item-id="item-1761042420910-ila6vr3ic" data-list-item-path="item-1761042420910-ila6vr3ic"><div class="list-item-paragraph">Find $\overrightarrow{BC}$ in terms of $\mathbf{a}$ and $\mathbf{b}$. Hence show that&nbsp;$AB$ is perpendicular to $BC$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1761042421443-l9jlig7ts" data-list-item-path="item-1761042421443-l9jlig7ts"><div class="list-item-paragraph">Show that the area of triangle $ABC$ can be written as&nbsp;$k\left| \mathbf{a}\times \mathbf{b} \right|$ where&nbsp;$k$ is a constant to be found. Hence find, in terms of $\left| \mathbf{a} \right|$, the maximum area of triangle $ABC$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1761042422065-6fjlaxp0i" data-list-item-path="item-1761042422065-6fjlaxp0i"><div class="list-item-paragraph">Given that&nbsp;$\measuredangle AOB=30{}^\circ $, find $\overrightarrow{OF}$ where&nbsp;$F$ is the foot of perpendicular of $B$ to $AC$. Hence, find $\overrightarrow{OB'}$ where&nbsp;$B'$ is the reflection of $B$ along the line $AC$.<span class="inline-marks">[3]</span></div></li></ol>

The points $A$, $B$ and $C$ lie on a circle with center $O$ and diameter $AC$. It is given that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$.

Perpendicular Projection and Reflection in a Line

2013 AJC P2 Q2

<p>Two points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively relative to origin $O$.</p>
<p>The points $P$ on $OA$ and $Q$ on $AB$ are such that&nbsp;$OP=3PA$ and $5AQ=7QB$.</p>
<p>The point $R$ is on $PQ$ produced so that&nbsp;$5PQ=4QR$.</p>
<p>Express $\overrightarrow{AQ}$, $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.</p>
<p>Prove by a vector method that&nbsp;$O$, $B$, $R$ are collinear.<span class="inline-marks">[6]</span></p>

Two points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively relative to origin $O$.

The points $P$ on $OA$ and $Q$ on $AB$ are such that $OP=3PA$ and $5AQ=7QB$.

The point $R$ is on $PQ$ produced so that $5PQ=4QR$.

Express $\overrightarrow{AQ}$, $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

Prove by a vector method that $O$, $B$, $R$ are collinear.

JPJC Vectors Tutorial 1(a) Q3

<p>The points ${P}$ and ${Q}$ have position vectors ${\mathbf{a} + \mathbf{b}}$ and ${3\mathbf{a} - 2\mathbf{b}}$ respectively relative to the origin ${O}$.</p>
<p>Given that&nbsp;${OPQR}$ is a parallelogram, express the vectors ${\overrightarrow{PQ}}$ and ${\overrightarrow{PR}}$ in terms of ${\mathbf{a}}$ and ${\mathbf{b}}$.</p>
<p>By evaluating two scalar products, show that if&nbsp;${OPQR}$ is a square, then&nbsp;${\left| \mathbf{a} \right|^2 = 2\left| \mathbf{b} \right|^2}$.</p>

The points ${P}$ and ${Q}$ have position vectors ${\mathbf{a} + \mathbf{b}}$ and ${3\mathbf{a} - 2\mathbf{b}}$ respectively relative to the origin ${O}$.

Given that ${OPQR}$ is a parallelogram, express the vectors ${\overrightarrow{PQ}}$ and ${\overrightarrow{PR}}$ in terms of ${\mathbf{a}}$ and ${\mathbf{b}}$.

By evaluating two scalar products, show that if ${OPQR}$ is a square, then ${\left| \mathbf{a} \right|^2 = 2\left| \mathbf{b} \right|^2}$.

RI Vectors Tutorial Q9

<p>Let the $A$, $B$ and $C$ be points on the same plane with position vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively and $\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}$. It is given that vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are unit vectors.</p>
<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">By considering $\mathbf{c}\cdot \mathbf{c}$, find the value of $\mathbf{a}\cdot \mathbf{b}$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the angle $\angle AOB$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Draw the position vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ on a single diagram. Using your diagram, identify the type of triangle $ABC$.</div><p style="text-align: right;">[2]</p></li></ol></li><li><div class="list-item-paragraph">The point $D$ has position vector $\mathbf{a}+\mathbf{b}$. Find the area of the quadrilateral $ACBD$.</div><p style="text-align: right;">[2]</p></li></ol>

Let the $A$, $B$ and $C$ be points on the same plane with position vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively and $\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}$. It is given that vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are unit vectors.

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