CAIE A-Level Mathematics A2 3.7.3 Position Vectors Questions

State $A B=\sqrt{7^{2}+1^{2}+4^{2}}(=\sqrt{66})$
Or $(A B)^{2}=66$ Condone a sign error in $\overrightarrow{A B}$.

State $\overrightarrow{B D}$ in component form
$\left(\begin{array}{c}-7+3 r -1+r 4-2 r\end{array}\right)$ or equivalent.

$$\begin{aligned} A B=B D \Rightarrow(3 r-7)^{2}+(r-1)^{2}+(-2 r+4)^{2}=66 \left(14 r^{2}-60 r=0\right) \end{aligned}$$
Or equivalent equation in one unknown for their A B and their $\overrightarrow{B D} \neq \overrightarrow{O D}$.

If you never see a correct form and they go direct to $9 r^{2}+49+r^{2}+1 \ldots .$. then M0.

$\Rightarrow r=\frac{30}{7}$
Correct only. Ignore r=0 if seen.

$\overrightarrow{O D}=\frac{76}{7} \mathbf{i}+\frac{37}{7} \mathbf{j}-\frac{32}{7} \mathbf{k}$
Must be a vector.

Condone if also have $\overrightarrow{O D}=\overrightarrow{O A}$.