CAIE A-Level Mathematics 3.7 Vectors Question Bank

Obtain direction vector -i+j+2 k, or equivalent
B1
Accept answers as column vectors throughout.
Use a correct method to form a vector equation
M1
State answer $\mathbf{r}=\mathbf{i}+2 \mathbf{j}-\mathbf{k}+\lambda(-\mathbf{i}+\mathbf{j}+2 \mathbf{k})$, or equivalent correct form
A1
e.g. $\mathbf{r}=\left(\begin{array}{l}0 1\end{array}\right)+\mu\left(\begin{array}{c}1 -1 -2\end{array}\right)$ Allow $\left(\begin{array}{l}x y z\end{array}\right)$ for r.

Use a correct method to find the position vector of C
M1
e.g. $\mathbf{O C}=\mathbf{O A}+\mathbf{A C}=\left(\begin{array}{c}1-3 2+3 -1+6\end{array}\right)$
Obtain answer -2 i+5 j+5 k, or equivalent
A1
Accept as coordinates.

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.