Review question

# If $Z$ is on $OA$ and also on $BY$, what is its position vector? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8798

## Question

$O$ is the origin, $\overrightarrow{OA} = \mathbf{a}$, and $\overrightarrow{OB} = \mathbf{b}$. The point $X$ is on $AB$ such that $AX = 2XB$ and $Y$ is the midpoint of $OX$.

Find $\overrightarrow{OX}$ and hence show that $\overrightarrow{BY} = \dfrac{1}{6}\mathbf{a}-\dfrac{2}{3}\mathbf{b}$.

$BY$ produced meets $OA$ at $Z$. Using the facts that $\overrightarrow{OZ} = \mathbf{b} + k\,\overrightarrow{BY} \quad \text{for some value of } k$ and $\overrightarrow{OZ} = h\,\mathbf{a} \quad \text{for some value of } h,$ find the position vector of $Z$.