Let two non-parallel vectors and have magnitudes and

MATHEMATICS · Subjective · A Level · Mathematics · Vectors · Vectors

Problem

Let two non-parallel vectors

\mathbf{A}

and

\mathbf{B}

have magnitudes

A

and

B

, respectively, with the angle between them being

\theta

(where

0<\theta<\pi

). (a) Write an expression for the vector sum

\mathbf{R}=\mathbf{A}+\mathbf{B}

in terms of

A

B

, and

\theta

. (b) Derive an expression for the magnitude

R=|\mathbf{R}|

. (c) Hence obtain a simplified symbolic expression for the unit vector

\hat{r}

along

\mathbf{R}

in terms of

A

B

\theta

, and the unit vectors along

\mathbf{A}

and

\mathbf{B}

. (d) Show that when

A=B

, the direction of

\hat{r}

bisects the angle between

\mathbf{A}

and

\mathbf{B}