CAIE IGCSE Additional Mathematics 13.3 Vector Calculations Question Bank

Practise combining vector paths and expressing points on lines in terms of base vectors and scalar parameters.

Exam points

write a point on a line as a+k(b−a) before simplifying in the base vectors
equate coefficients of independent vectors to solve for unknown scalars
substitute the parameter value back to obtain the required position vector

[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
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 $8\binom{2}{-3}$ .

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13.3. Calculate and combine vectors question 1