Question 1216

Vectors I: Basics, Dot and Cross Products

2016 Paper 1 Question 5 Variant

Question

The vectors

{\mathbf{u}}

and

{\mathbf{v}}

are given by

{\mathbf{u} = - 4 \mathbf{i} + 5 \mathbf{j} - 4 \mathbf{k}}

and

{\mathbf{v} = a\mathbf{i} + b \mathbf{k},}

where

{a}

and

{b}

are constants.

(i)

Find

{(\mathbf{u} + \mathbf{v})\times(\mathbf{u} - \mathbf{v})}

in terms of

{a}

and

{b.}

[2]

(ii)

Given that the

{\mathbf{i}\textrm{-}}

and

{\mathbf{k}\textrm{-}}

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 exact form, the possible values of

{a}

for which

{(\mathbf{u} + \mathbf{v})\times(\mathbf{u} - \mathbf{v})}

is a unit vector.

[4]

(iii)

Given instead that

{(\mathbf{u} + \mathbf{v})\cdot(\mathbf{u} - \mathbf{v})=0,}

find the numerical value of

{|\mathbf{v}|.}

[2]