N2016 P1 Q5

The vectors \(\mathbf u\) and \(\mathbf v\) are given by \(\mathbf u=2\mathbf i-\mathbf j+2\mathbf k\) and \(\mathbf v=a\mathbf i+b\mathbf k\), where \(a\) and \(b\) are constants.

1. Find \((\mathbf u+\mathbf v)\times(\mathbf u-\mathbf v)\) in terms of \(a\) and \(b\).[2]
2. Given that the \(\mathbf i\)- and \(\mathbf k\)-components of the answer to part (i) are equal, express \((\mathbf u+\mathbf v)\times(\mathbf u-\mathbf v)\) in terms of \(a\) only. Hence find, in an exact form, the possible values of \(a\) for which \((\mathbf u+\mathbf v)\times(\mathbf u-\mathbf v)\) is a unit vector.[4]
3. Given instead that \((\mathbf u+\mathbf v)\cdot(\mathbf u-\mathbf v)=0\), find the numerical value of \(|\mathbf v|\).[2]