Question 1218

Vectors I: Basics, Dot and Cross Products

2018 Paper 1 Question 6 Variant

Question

Vectors

{\mathbf{a}, \mathbf{b}}

and

{\mathbf{c}}

are such that

{\mathbf{a} \neq \mathbf{0}}

and

{\mathbf{a} \times 4 \mathbf{b} = 5 \mathbf{a} \times \mathbf{c}}

(i)

Show that

{4 \mathbf{b}-5 \mathbf{c}=\lambda \mathbf{a}, }

where

{\lambda}

is a constant.

[2]

(ii)

It is now given that

{\mathbf{a}}

and

{\mathbf{c}}

are unit vectors, that the modulus of

{\mathbf{b}}

{3}

and the angle between

{\mathbf{b}}

and

{\mathbf{c}}

{120^\circ}

Using a suitable scalar product, find exactly the two possible values of

{\lambda.}

[5]