CAIE IGCSE Additional Math 13 vectors in two dimensions

Use this Vectors hub to connect vector routes, component notation, unit vectors, collinearity, ratios and velocity-position modelling.

Covered syllabus points

13.1. Use vector notation 13.2. Use position and unit vectors 13.3. Calculate and combine vectors 13.4. Compose and resolve velocities

[Maximum number: 7]

The point O is the origin.
Two points P and Q are such that $\overrightarrow{PQ}$ is in the same direction as $-\mathbf{i}+5\mathbf{j}$ .

(a)

The point R is such that $\overrightarrow{OR}$ is in the same direction as $\overrightarrow{PQ}$ and the magnitude of $\overrightarrow{OR}$ is $3\sqrt{26}$ .
Find $\overrightarrow{OR}$ .

[ 3 ]

$[\overrightarrow{OR}=]k(-\mathbf{i}+5\mathbf{j})$ or $k\binom{-1}{5}$
B1 soi
$\sqrt{1^2+5^2}=\sqrt{26}$
B1 for finding the magnitude of $\overrightarrow{PQ}$ ; allow unsimplified.
$-3\mathbf{i}+15\mathbf{j}$ or $\binom{-3}{15}$
B1
Allow $3(-\mathbf{i}+5\mathbf{j})$ or $3\binom{-1}{5}$ . Do not isw.

(b)

$\overrightarrow{OP}$ is in the same direction as $2\mathbf{i}-3\mathbf{j}$ and $\overrightarrow{OQ}=10\mathbf{i}+6\mathbf{j}$ .
Find $\overrightarrow{OP}$ .

[ 4 ]

$a(2\mathbf{i}-3\mathbf{j})+b(-\mathbf{i}+5\mathbf{j})=10\mathbf{i}+6\mathbf{j}$ oe
B1
2a-b=10 oe, -3a+5b=6 oe
or $\frac{10-2a}{6+3a}=\frac{-1}{5}$ or $\frac{10+b}{6-5b}=\frac{-2}{3}$
M1 dep on previous B1 for equating like vectors.
a=8 or b=6
A1
$16\mathbf{i}-24\mathbf{j}$ oe
A1
Allow $8(2\mathbf{i}-3\mathbf{j})$ or $8\binom{2}{-3}$ .

[Maximum number: 7]

(a)

In this question, $\mathbf{i}$ is a unit vector due east and $\mathbf{j}$ is a unit vector due north.
A cyclist rides at a speed of $4\,\mathrm{m\,s^{-1}}$ on a bearing of $015^\circ$ . Write the velocity vector of the cyclist in the form $x\mathbf{i}+y\mathbf{j}$ , where x and y are constants.

[ 2 ]

$4\cos75^\circ\,\mathbf{i}+4\sin75^\circ\,\mathbf{j}$
or $4\sin15^\circ\,\mathbf{i}+4\cos15^\circ\,\mathbf{j}$ oe, isw
B2
B1 for $x=4\cos75^\circ$ or $x=4\sin15^\circ$ oe, soi, or $y=4\sin75^\circ$ or $y=4\cos15^\circ$ oe, soi.
SC1 for a correct expression with missing brackets such as $\sqrt6-\sqrt2\,\mathbf{i}+\sqrt6+\sqrt2\,\mathbf{j}$ , or for (1.04,3.86) oe.

(b)

A vector of magnitude 6 on a bearing of $300^\circ$ is added to a vector of magnitude 2 on a bearing of $230^\circ$ to give a vector $\mathbf{v}$ . Find the magnitude and bearing of $\mathbf{v}$ .

[ 5 ]

$(-6\cos30^\circ-2\cos40^\circ)\mathbf{i}+(6\sin30^\circ-2\sin40^\circ)\mathbf{j}$ oe, soi
B1
$r^2=(-6.7282\ldots)^2+(1.7144\ldots)^2$
M1
r=6.94
or $6.9432329\ldots$ rot to 4 or more sf
A1
$\alpha=\tan^{-1}\left(\frac{6.7282\ldots}{1.7144\ldots}\right)$ or awrt 75.7
M1
Bearing $284^\circ$ or $284.2^\circ$ rot to 4 or more sf
A1

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Covered syllabus points

13. Vectors in two dimensions question 1

Question (a)

Question (b)

13. Vectors in two dimensions question 2

Question (a)

Question (b)