In a parallelogram

MATHEMATICS · Multi-part · A Level · Mathematics · Vectors · Vectors

Problem

In a parallelogram

OABC

, the adjacent sides are represented by vectors

\mathbf{a}=\overrightarrow{OA}

and

\mathbf{b}=\overrightarrow{OC}

. (i) Express the diagonals

\overrightarrow{OB}

and

\overrightarrow{AC}

in terms of

\mathbf{a}

and

\mathbf{b}

. (ii) Show that the diagonals bisect each other by finding the position vector of their point of intersection. (iii) If

|\mathbf{a}|=7

units,

|\mathbf{b}|=5

units and

\mathbf{a}\cdot\mathbf{b}=17

, calculate the lengths of the two diagonals.