Q16 (a) (iii)

Criteria	Marks
Provides correct solution	2
Chooses one suitable vector to use with the result from part (ii), or equivalent merit	1

Sample answer:

If $P(x,y,z)$ is on the unit sphere, $x^2+y^2+z^2=1$, hence $Q(|x|,|y|,|z|)$ is on the unit sphere.

In part (ii), let
$\mathbf a=\begin{pmatrix}|x|\\|y|\\|z|\end{pmatrix},\qquad \mathbf b=\begin{pmatrix}1\\1\\1\end{pmatrix}.$

Then
$|x|+|y|+|z|\le\sqrt{|x|^2+|y|^2+|z|^2}\sqrt{1^2+1^2+1^2}=\sqrt3.$

$\therefore |x|+|y|+|z|\le\sqrt3.$