Consider the vectors ,

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

Problem

Consider the vectors

\mathbf{a}=2\,\hat{i}-\hat{j}+3\,\hat{k}

\mathbf{b}=4\,\hat{i}+5\,\hat{j}-2\,\hat{k}

, and

\mathbf{c}=\,\hat{i}+2\,\hat{j}+\hat{k}

. A parallelogram is to be constructed whose one side is the vector

\mathbf{b}

and the adjacent side is

\mathbf{a}'=\mathbf{a}+k\,\mathbf{c}

, where

k

is a scalar parameter. Answer the following: (a) Obtain an expression for

\mathbf{a}'\times\mathbf{b}

, and hence write the area of the parallelogram in terms of

k

. Do not evaluate

k

at this stage. (b) If the area of the triangle formed by the vectors

\mathbf{a}'

and

\mathbf{b}

is exactly

6

square units, determine all admissible value(s) of

k

. (Assume

k

is real.)