Question 1215

Vectors I: Basics, Dot and Cross Products

2015 Paper 1 Question 7 Variant

Question

Referred to the origin

{O,}

points

{A}

and

{B}

have position vectors

{\mathbf{a}}

and

{\mathbf{b}}

respectively.

Point

{C}

lies on

{OA,}

between

{O}

and

{A,}

such that

{OC:CA = 4:1.}

Point

{D}

lies on

{OB,}

between

{O}

and

{B,}

such that

{OD:DB = 1:1.}

(i)

Find the position vectors

{\overrightarrow{OC}}

and

{\overrightarrow{OD},}

giving your answers in terms of

{\mathbf{a}}

and

{\mathbf{b}.}

[2]

(ii)

Show that the vector equation of the line

{BC}

can be written as

\mathbf{r} = \frac{4}{5} \lambda \mathbf{a} + (1-\lambda) \mathbf{b},

where

{\lambda}

is a parameter.

Find in a similar form the vector equation of the line

{AD}

in terms of a parameter

{\mu.}

[3]

(iii)

Find, in terms of

{\mathbf{a}}

and

{\mathbf{b},}

the position vector of the point

{E}

where the lines

{BC}

and

{AD}

meet and find the ratio

{AE:ED.}

[5]