Question 1213

Vectors I: Basics, Dot and Cross Products

2013 Paper 1 Question 6 Variant

Question

The origin

{O}

and the points

{A, B}

and

{C}

lie in the same plane, where

{\overrightarrow{OA}=\mathbf{a}, }

{\overrightarrow{OB}=\mathbf{b} }

and

{\overrightarrow{OC}=\mathbf{c}.}

The point

{N}

is on

{AC}

such that

{AN:NC = 1:1.}

(i)

Write down the position vector of

{N}

in terms of

{\mathbf{a}}

and

{\mathbf{c}.}

[1]

(ii)

It is given that

{\mathbf{c}=\lambda\mathbf{a}+\mu\mathbf{b},}

for positive constants

{\lambda}

and

{\mu.}

It also given that the area of triangle

{ONC}

is equal to the area of triangle

{OMC,}

where

{M}

is the mid-point of

{OB.}

By finding the areas of these triangles in terms of

{\mathbf{a}}

and

{\mathbf{b},}

find

{\lambda}

in terms of

{\mu.}

[5]