Let and be two planar vectors

MATHEMATICS · Multi-part · CBSE · Mathematics · Vector Algebra · Vector Algebra

Problem

Let

\mathbf{A} = (k+1)\hat{i} + 3\hat{j}

and

\mathbf{B} = (4 - k)\hat{i} + (k - 2)\hat{j}

be two planar vectors, where

k

is a real parameter. The resultant vector is

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

. (i) Write down the

x

- and

y

-components of

\mathbf{R}

in terms of

k

. (ii) For what value(s) of

k

is the magnitude of

\mathbf{R}

equal to

5

? (iii) For the value(s) of

k

obtained in part (ii), determine whether

\mathbf{R}

makes an angle of

45^{\circ}

with the positive

x

-axis. Justify your answer.