Review question

# What's the position vector of the reflection of $O$ in the plane $ABC$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9794

## Suggestion

The position vectors of the points $A, B$ and $C$ with respect to an origin $O$ are the unit vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ respectively. It is given that $OB$ is perpendicular to $OC$, $m = \cos \widehat{AOC}$, $n = \cos \widehat{AOB}$, and $m+n \neq 1$. The vector $\mathbf{r}$, where $\mathbf{r} = \mathbf{a} + \lambda \mathbf{b} + \mu \mathbf{c}$, is perpendicular to the plane $ABC$. Show that

$\lambda = \dfrac{(1-m)(1+m-n)}{1-m-n}$

and find a similar expression for $\mu$.

Note that $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are unit vectors and that two of them are perpendicular.

If $\mathbf{r}$ is perpendicular to the plane $ABC$, then which vectors is $\mathbf{r}$ perpendicular to?

If two vectors are perpendicular, what can we write down algebraically?

It is further given that $\widehat{AOB}=\widehat{AOC}=120^\circ$. The point $N$ is the foot of the perpendicular from $O$ on the plane $ABC$. Find the position vector of $N$, and deduce that the position vector of the reflection of $O$ in the plane $ABC$ is

$\dfrac{4}{5}\mathbf{a}+ \dfrac{3}{5}\mathbf{b} + \dfrac{3}{5}\mathbf{c}.$

Can we use our previous results to find $\mathbf{n}$, the position vector of $N$? If we know $\mathbf{n}$, can we reflect $O$ in $N$ easily?