1. ## Vector Help

I dont know how to do qn 7 and 8. pls help pls.Ignore the blue ink on the diagram.

2. 7)
a) i) $\displaystyle \overrightarrow{OD}=\frac{4}{5}\overrightarrow{OM} =\frac{4}{5}(\overrightarrow{OA}+\overrightarrow{A B})=$

$\displaystyle =\frac{4}{5}\left(\overrightarrow{OA}+\frac{1}{2}\ overrightarrow{AB}\right)=\frac{4}{5}\left(\overri ghtarrow{OA}+\frac{1}{2}\overrightarrow{OC}\right) =$

$\displaystyle =\frac{4}{5}\left(p+q+\frac{p-q}{2}\right)=\frac{2(3p+q)}{5}$

ii) $\displaystyle \overrightarrow{DB}=\overrightarrow{DM}+\overright arrow{MB}=\frac{1}{5}\overrightarrow{OM}+\frac{1}{ 2}\overrightarrow{AB}=$

$\displaystyle =\frac{1}{5}\left(\overrightarrow{OA}+\frac{1}{2}\ overrightarrow{OC}\right)+\frac{1}{2}\overrightarr ow{OC}=$

$\displaystyle =\frac{1}{5}+\left(p+q+\frac{p-q}{2}\right)+\frac{p-q}{2}=\frac{4p-2q}{5}$

iii) $\displaystyle \overrightarrow{NB}=\overrightarrow{NA}+\overright arrow{AB}=\frac{1}{3}\overrightarrow{OA}+\overrigh tarrow{OC}=\frac{4p-2q}{3}$

b) $\displaystyle \overrightarrow{DB}=\frac{3}{5}\overrightarrow{NB}$

$\displaystyle \overrightarrow{DB}, \ \overrightarrow{NB}$ are colinear.

3. 8) $\displaystyle \overrightarrow{OA}=\begin{pmatrix}p\\0\end{pmatri x}, \ \overrightarrow{OC}=\begin{pmatrix}q\\r\end{pmatri x}$

$\displaystyle \overrightarrow{OB}=\overrightarrow{OA}+\overright arrow{OC}=\begin{pmatrix}p+q\\r\end{pmatrix}$

$\displaystyle \overrightarrow{AC}=\overrightarrow{OC}-\overrightarrow{OA}=\begin{pmatrix}p-q\\r\end{pmatrix}$

$\displaystyle OB^2+AC^2=(p+q)^2+r^2+(q-p)^2+r^2=2(p^2+q^2+r^2)$

$\displaystyle 2(OA^2+OC^2)=2(q^2+r^2+r^2)$

Then, $\displaystyle OB^2+AC^2=2(OA^2+OC^2)$