If $x^2 + y^2 + z^2 = 1$
Maximise $(cy-bz)^2 + (az-cx)^2 + (bx-ay)^2$
This can be written as the square of the magnitude of the determinant:
$$ \begin{vmatrix} \widehat{i} & \widehat{j} & \widehat{k} \\ x & y & z \\ a & b & c \\ \end{vmatrix} $$
which is the cross product of $(x,y,z)$ and $(a,b,c)$ hence the maximum is $a^2+b^2+c^2$
How do we find the maximum of a general expression: $\sum_{i=1}^3(a_{i}x+b_{i}y+c_{i}z)^2$
What about the special case: $c_{3} = b_{2} = a_{1} = 0$
and an even more special case: $p^2(cy-bz)^2 + q^2(az-cx)^2 + r^2(bx-ay)^2$