CAIE A-Level Mathematics 3.7.3 Position vectors

Practise finding position vectors, displacement vectors, magnitudes and unit vectors from 2D or 3D coordinate information.

Exam points

calculate vector magnitudes to compare lengths or prove triangles are isosceles
form position vectors for points on a line, then solve modulus conditions for the parameter
find unit vectors by dividing a direction vector by its magnitude

[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
M1
Simplify and obtain $3 \lambda^{2}-\lambda-4=0$ , or equivalent
A1
$3 \lambda^{2}+\lambda-4=0$ 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
M1
$\left(\lambda=-1, \frac{4}{3}\right.$ 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
A1
Accept as coordinates.

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3.7.3—Position vectors question 1