Q1
Answer
For the following vectors, find their magnitude and the angle they make with the positive $x$ axis
  1. $\mathbf{i} + 4\mathbf{j}$
  2. $-3\mathbf{i} - 2\mathbf{j}$
  1. Magnitude $\sqrt{17}$, angle: $76.0^{\circ}$
  2. Magnitude $\sqrt{13}$, angle: $-146^{\circ}$
Q2
Answer
Given that $(3a+b)\mathbf{i} + \mathbf{j} + ac\mathbf{k} = 7\mathbf{i} - b\mathbf{j} + 4\mathbf{k}$, find the values of $a$, $b$ and $c$.
$a = \dfrac{8}{3}$, $b = -1$, $c = \dfrac{3}{2}$
Q3
Answer
The points $A$, $B$ and $C$ have position vectors $\begin{pmatrix}1\\0\\0\end{pmatrix}$, $\begin{pmatrix}5\\3\\4\end{pmatrix}$, and $\begin{pmatrix}2\\-1\\8\end{pmatrix}$.
  1. Determine if triangle $ABC$ is isosceles.
  2. Find the area of triangle $ABC$.
  1. $|AB| = |BC|$ only, isosceles
  2. $\dfrac{7}{2}\sqrt{33}$
Q4
Answer
Given that $\mathbf{a} = 3\mathbf{i} + 4\mathbf{j}$ and $\mathbf{b} = \mathbf{i} - 2\mathbf{j}$, find:
  1. $\lambda$ such that $\mathbf{a} + \lambda\mathbf{b}$ is parallel to $\mathbf{i} - \mathbf{j}$
  2. $\mu$ such that $\mathbf{a} + \mu\mathbf{b}$ is parallel to $3\mathbf{i} + 4\mathbf{j}$.
  1. $\lambda = 7$
  2. $\mu = 0$
Q5
Answer
Given that $\mathbf{a} = \begin{pmatrix}p\\-q\end{pmatrix}$, $\mathbf{b} = \begin{pmatrix}q\\p\end{pmatrix}$, and $\mathbf{c} = \begin{pmatrix}7\\4\end{pmatrix}$, find $p$ and $q$ if $2\mathbf{a} + \mathbf{b} = \mathbf{c}$.
$p = 3.6$ and $q = -0.2$
Q6
Answer
Given $\mathbf{a} = 3\mathbf{i} - 2\mathbf{j}$, $\mathbf{b} = p\mathbf{i} - 2p\mathbf{j}$ and that the resultant is parallel to the vector $5\mathbf{i} - 8\mathbf{j}$. Find the resultant vector.
$p = 7$, resultant: $10\mathbf{i} - 16\mathbf{j}$
Q7
Answer
Given that $\mathbf{a} = p\mathbf{i} + q\mathbf{j}$ has $|\mathbf{a}| = 10$ and it makes an angle $\theta$ with the positive $x$ axis such that $\sin\theta = \dfrac{3}{5}$, find the possible values of $p$ and $q$.
$p = 8$, $q = 6$
Q8
Answer
The point $A$ lies on the circle $x^2 + y^2 = 9$. Given that $\overrightarrow{OA} = 2k\mathbf{i} + k\mathbf{j}$, find the exact values $k$ could be.
$k = \pm\dfrac{3\sqrt{5}}{5}$
Q9
Answer
In the following diagram, $P$, $Q$ and $R$ are the midpoints of the sides shown. Given that $\overrightarrow{OP} = \mathbf{p}$ and $\overrightarrow{OR} = \mathbf{r}$, show that triangles $OAB$ and $PQR$ are similar.
$\overrightarrow{BQ} = \mathbf{p}-\mathbf{r}$ <br> $\overrightarrow{OB} = 2\mathbf{r}$ and $\overrightarrow{PQ} = \mathbf{r}$ so these are parallel and the angles are all the same
Q10
Answer
In the following diagram of a trapezium, $OB$ is parallel to $AC$, and $BD:DA=1:2$. Given that $\overrightarrow{OA} = 4\mathbf{a}$, $\overrightarrow{OB} = 3\mathbf{b}$ and $\overrightarrow{AC} = 6\mathbf{b}$, show that $ODC$ is a straight line and find the ratio $OD:DC$.
$\overrightarrow{OC} = 4\mathbf{a} + 6\mathbf{b}$, $\overrightarrow{OD} = \frac{1}{3}\overrightarrow{OC}$. <br> Ratio $1:2$, parallel and share a point.
Q11
Answer
$\overrightarrow{OA}$ is the vector $4\mathbf{i} - \mathbf{j} - 2\mathbf{k}$, and $\overrightarrow{OB}$ is the vector $-2\mathbf{i} + 3\mathbf{j} + \mathbf{k}$. Find the unit vector in the direction of $\overrightarrow{AB}$.
$\dfrac{1}{\sqrt{61}}(-6\mathbf{i} + 4\mathbf{j} + 3\mathbf{k})$
Q12
Answer
$A$, $B$, and $C$ have vectors $\begin{pmatrix}8\\-7\\4\end{pmatrix}$, $\begin{pmatrix}8\\-3\\3\end{pmatrix}$, and $\begin{pmatrix}12\\-6\\3\end{pmatrix}$
  1. Determine if triangle $ABC$ is isosceles.
  2. Find the area of triangle $ABC$.
  1. $|AB| = \sqrt{17}$, $|AC| = \sqrt{18}$, $|BC| = 5$. Scalene.
  2. $8.38$
Q13
Answer
$OAB$ is a triangle. $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. The point $M$ divides $OA$ in the ratio $2:1$. $MN$ is parallel to $OB$. Show that $AN:NB = 1:2$.
$\overrightarrow{ON} = \frac{2}{3}\mathbf{a} + \frac{1}{3}\mathbf{b}$ <br> $\overrightarrow{AN} = \frac{1}{3}(\mathbf{b}-\mathbf{a})$ <br> $\overrightarrow{NB} = \frac{2}{3}(\mathbf{b}-\mathbf{a})$ so ratio is $1:2$
Q14
Answer
$\overrightarrow{PQ} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}$, and $\overrightarrow{QR} = -2\mathbf{i} + 4\mathbf{j} + 3\mathbf{k}$
  1. Find angle $PQR$.
  2. Find the area of triangle $PQR$.
  1. $78.5^{\circ}$
  2. $9.87$
Q15
Answer
$OABC$ is a square. $M$ is the midpoint of $OA$ and $Q$ divides $BC$ in the ratio $1:3$. $AC$ and $MQ$ meet at the point $P$. By writing $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OC} = \mathbf{c}$, show that $P$ divides $AC$ in the ratio $2:3$.
$\overrightarrow{OM} = \frac{1}{2}\mathbf{a}$ <br> $\overrightarrow{MQ} = \frac{1}{4}\mathbf{a} + \mathbf{c}$ <br> $\overrightarrow{AC} = \mathbf{c} - \mathbf{a}$ <br> $\overrightarrow{OP} = \frac{3}{5}\mathbf{a} + \frac{2}{5}\mathbf{c}$ <br> $\overrightarrow{AP} = \frac{2}{5}(\mathbf{c} - \mathbf{a})$ and $\overrightarrow{PC} = \frac{3}{5}(\mathbf{c} - \mathbf{a})$
Q16
Answer
$\overrightarrow{AB} = 7\mathbf{i} - \mathbf{j} + 2\mathbf{k}$, and $\overrightarrow{BC} = -\mathbf{i} + 5\mathbf{k}$. The point $D$ is such that $\overrightarrow{AD} = 3\overrightarrow{AB}$, and the point $E$ such that $\overrightarrow{AE} = 3\overrightarrow{AC}$.
  1. Find the area of triangle $ABC$.
  2. Find the area of triangle $ADE$.
  1. $18.7$
  2. $3^2\times 18.7... = 168$
Q17
Answer
Given the points $A$, $B$, $C$ and $D$ have coordinates $(2,-5,-8)$, $(1,-7,-3)$, $(0,15,-10)$ and $(2,19,-20)$ respectively, show that $ABCD$ is a trapezium.
$AB = -0.5CD$ so they are parallel. No other sides are parallel, so trapezium.
Q18
Answer
$P$, $Q$ and $R$ have coordinates $(4,-9,-3)$, $(7,-7,-7)$ and $(8,-2,0)$. Find the coordinates of $S$ such that $PQRS$ is a parallelogram.
$(5,-4,4)$
Q19
Answer
Given $A$, $B$, $C$ and $D$ have coordinates $(7,12,-1)$, $(11,2,-9)$, $(14,-14,3)$ and $(8,1,15)$ respectively, describe the quadrilateral $ABCD$.
$CD = -1.5AB$, 1 pair of parallel sides so trapezium.
Q20
Answer
$OABC$ is a parallelogram and the point $M$ is the midpoint of $AB$. The point $N$ lies on the diagonal $AC$ such that $AN:NC = 1:2$.
Given that $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OC} = \mathbf{c}$, show that $OMN$ is a straight line.
$\overrightarrow{ON} = \frac{2}{3}\mathbf{a} + \frac{1}{3}\mathbf{c}$ <br> $\overrightarrow{NM} = \frac{1}{2}\overrightarrow{ON}$. <br> Parallel and share a point, so straight line.
Q21
Answer
The points $A$ and $B$ have position vectors $10\mathbf{i} - 23\mathbf{j} + 10\mathbf{k}$ and $p\mathbf{i} + 14\mathbf{j} - 22\mathbf{k}$ respectively, relative to a fixed origin $O$. Given $OAB$ is an isosceles triangle, find three possible values for $p$.
$p = \pm7, \dfrac{1813}{20}$
Q22
Answer
The following diagram shows a trapezium where $AD$ is parallel to $BC$.
  1. Find $k$.
  2. Find and simplify an expression for $\overrightarrow{AB}$.
  1. $-6$
  2. $-\mathbf{a} - 7\mathbf{b}$
Q23
Answer
In the following diagram, $OB:BE = 1:2$, $OC:CA = 1:2$ and $BD:DA = 1:3$. Given that $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$,
  1. Show that $CDE$ is a straight line, and find the ratio $CD:DE$.
  2. Show that $BC$ is parallel to $EA$, and find the ratio $BC:EA$.
  1. $\overrightarrow{CD} = -\frac{1}{12}\mathbf{a} + \frac{3}{4}\mathbf{b}$, $\overrightarrow{DE} = 3\overrightarrow{CD}$ ratio: $1:3$
  2. $\overrightarrow{BC} = \frac{1}{3}\mathbf{a} - \mathbf{b}$, $\overrightarrow{EA} = 3\overrightarrow{BC}$ ratio: $1:3$
Q24
Answer
Given that $\overrightarrow{OA} = \mathbf{a}$, $\overrightarrow{OB} = \mathbf{b}$, $\overrightarrow{OC} = \mathbf{2a}$ and $\overrightarrow{OD} = 2\mathbf{a} + \mathbf{b}$, show that $E$ lies on the line $AB$ if $\overrightarrow{OE} = \frac{1}{3}\overrightarrow{OD}$.
$\overrightarrow{AE} = \frac{1}{3}(\mathbf{b} - \mathbf{a})$ and $\overrightarrow{EB} = 2\overrightarrow{AE}$ so they are parallel and share a point.