CAIE A-Level Mathematics A2 3.7.2 Vector Operations Questions

Find $\overrightarrow{P Q}$ (or $\overrightarrow{Q P}$ ) for a general point Q on m
$$= \pm((1+2 \mu, 2-3 \mu, 3+\mu)-(a+\lambda c, 3-2 \lambda, b+4 \lambda))$$$$\left[\overrightarrow{P Q} \text { or } \overrightarrow{Q P}= \pm\left(\begin{array}{c} -3+2 \mu -5-3 \mu 5+\mu \end{array}\right)\right]$$

Could be their a, b, c and $\lambda$ values provided M1 M1 gained in (a). Allow expression in answer column.

Equate the scalar product of $\overrightarrow{P Q}$ (or $\overrightarrow{Q P}$ ) and a direction vector for m to
zero and obtain an equation in $\mu$

M1*

$$\begin{aligned} (2(-3+2 \mu)-3(-5-3 \mu)+(5+\mu))=0 \text { Allow } \overrightarrow{P Q}=\overrightarrow{O Q}+\overrightarrow{O P} \text { sign problem. } \end{aligned}$$

Solve and obtain $\mu=-1$$$\begin{aligned} P Q^{2}=(-3+2 \mu)^{2}+(-5-3 \mu)^{2}+(5+\mu)^{2} {\left[=14(\mu+1)^{2}+45\right] .} \text { Min when } \mu=-1 \text { or by differentiation. } \end{aligned}$$

Obtain $\overrightarrow{O Q}=-\mathbf{i}+5 \mathbf{j}+2 \mathbf{k}$ or $\overrightarrow{P Q}=-5 \mathbf{i}-2 \mathbf{j}+4 \mathbf{k}$

Must be labelled correctly
The working may be in (a) provided at least this result is used in (b).

Carry out a method to find the position vector of R

Alternative method for DM1
$$\begin{aligned} & \overrightarrow{OR}=(4,7,-2)+t(-5,-2,4) \overrightarrow{Q R}=\overrightarrow{OR}-\overrightarrow{O Q} & \text { Solve }|\boldsymbol{Q R}|^{2}=\frac{9}{4}|\boldsymbol{P Q}|^{2} \text { or }|\boldsymbol{Q R}|=\frac{3}{2}|\boldsymbol{P Q}| t=2.5 \end{aligned}$$
e.g. Use $\overrightarrow{OR}=\overrightarrow{O P}+\frac{5}{2} \overrightarrow{P Q}$ or $\overrightarrow{OR}=\overrightarrow{O Q}+\frac{3}{2} \overrightarrow{P Q}$ or $\overrightarrow{OR}=\frac{5}{2} \overrightarrow{O Q}-\frac{3}{2} \overrightarrow{O P}$ or $2 \overrightarrow{Q R}=2(\overrightarrow{OR}-\overrightarrow{O Q})=3 \overrightarrow{P Q}$ where $\overrightarrow{OR}=(x, y, z)$.

$\overrightarrow{P Q}$ used in all these approaches, may be incorrect, must be in the correct direction, i.e. not using $\overrightarrow{Q P}$ for $\overrightarrow{P Q}$.

Obtain $-\frac{17}{2} \boldsymbol{i}+2 \boldsymbol{j}+8 \boldsymbol{k}$ from correct working
Accept coordinates.
$$\text { Don't accept }-\frac{17}{2} \mathbf{i}+\frac{4}{2} \mathbf{j}+\frac{16}{2} \mathbf{k} \text {. }$$

SC2 Equate lines, attempt to find $\mu=-1$ or $\lambda=-1$ M1*
$\overrightarrow{O Q}=-\mathbf{i}+5 \mathbf{j}+2 \mathbf{k} \quad \mathrm{~A} 1$.
Attempt to find $\overrightarrow{O Q}$ using other parameter value DM1.
$\overrightarrow{O Q}=-\mathbf{i}+5 \mathbf{j}+2 \mathbf{k}$ therefore intersect A1.
Then use main scheme for the final DM1 A1.

First DM1 A1 are available if they show the 3
coordinates are consistent for the 2 parameter values
instead of attempting to find $\overrightarrow{O Q}$ using the other
parameter value and then showing intersection