CAIE IGCSE Additional Math 13.3 Calculate and Combine Vectors Questions

Practise solving two-dimensional vector geometry with position vectors, section ratios, collinearity proofs and exact length calculations in Cambridge International Additional Mathematics.

Exam points

Use position vectors, section ratios and vector equations to prove collinearity, find intersections and calculate lengths or bearings.

[Maximum number: 4]

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

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

$a(2\mathbf{i}-3\mathbf{j})+b(-\mathbf{i}+5\mathbf{j})=10\mathbf{i}+6\mathbf{j}$ oe
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 $16\mathbf{i}-24\mathbf{j}$ oe
Allow $8(2\mathbf{i}-3\mathbf{j})$ or $8\binom{2}{-3}$ .

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CAIE IGCSE Additional Math 13.3 Calculate and Combine Vectors Questions question 1