CAIE A-Level Mathematics A2 3.7.3 Position Vectors Questions

Practise using position vectors to represent points, lines and geometric relationships.

Exam points

represent points and displacements with position vectors
calculate lengths, midpoints and vector relationships
use position vectors to solve geometry and collinearity problems

[Maximum number: 5]

With respect to the origin O, the position vectors of the points A and B are given by $\overrightarrow{O A}=\left(\begin{array}{r}1 \\ 2 \\ -1\end{array}\right)$ and $\overrightarrow{O B}=\left(\begin{array}{l}0 \\ 3 \\ 1\end{array}\right)$ .

Find the possible position vectors of the point P on l such that $O P=\sqrt{14}$ .

State $\overrightarrow{O P}$ in component form
B1 FT
Form an equation in $\lambda$ by equating the modulus of O P to $\sqrt{14}$ , or equivalent
Simplify and obtain $3 \lambda^{2}-\lambda-4=0$ , or equivalent
$3 \lambda^{2}+\lambda-4=0$ if using i-j-2 k in (a).
$3 \mu^{2}+5 \mu-2=0$ if using -i+j+2 k in (a) and O B.
Solve a 3-term quadratic and find a position vector
$\left(\lambda=-1, \frac{4}{3}\right.$ or $\lambda=1,-\frac{4}{3}$ or $\mu=\frac{1}{3},-2$ or $\left.\mu=-\frac{1}{3}, 2\right)$
Obtain answers 2 i+j-3 k and $-\frac{1}{3} \mathbf{i}+\frac{10}{3} \mathbf{j}+\frac{5}{3} \mathbf{k}$ , or equivalent
Accept as coordinates.

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CAIE A-Level Mathematics A2 3.7.3 Position Vectors Questions question 1