$$\left( {\overrightarrow a \times \left( {\overrightarrow a \times \overrightarrow b } \right) + \overrightarrow b \times \left( {\overrightarrow a \times \overrightarrow b } \right)} \right) \times \left( {\overrightarrow a - \overrightarrow b } \right)$$

$$ = \left( {\overrightarrow a \left( {\overrightarrow a .\,\overrightarrow b } \right) - \overrightarrow b \left( {\overrightarrow a .\,\overrightarrow a } \right) + \overrightarrow a \left( {\overrightarrow b .\,\overrightarrow b } \right) - \overrightarrow b \left( {\overrightarrow a .\,\overrightarrow b } \right)} \right) \times \left( {\overrightarrow a - \overrightarrow b } \right)$$

$$ = \left( {\overrightarrow a .\,\overrightarrow b } \right)\left( {\overrightarrow a \times \overrightarrow a - \overrightarrow a \times \overrightarrow b } \right) - \left( {\overrightarrow a .\,\overrightarrow a } \right)\left( {\overrightarrow b \times \overrightarrow a - \overrightarrow b \times \overrightarrow b } \right) + \left( {\overrightarrow b .\,\overrightarrow b } \right)\left( {\overrightarrow a \times \overrightarrow a - \overrightarrow a \times \overrightarrow b } \right) - \left( {\overrightarrow a .\,\overrightarrow b } \right)\left( {\overrightarrow b \times \overrightarrow a - \overrightarrow b \times \overrightarrow b } \right)$$

$$ = \left( {\overrightarrow a .\,\overrightarrow b } \right)\left( {\overrightarrow b \times \overrightarrow a } \right) - \left( {\overrightarrow a .\,\overrightarrow a } \right)\left( {\overrightarrow b \times \overrightarrow a } \right) + \left( {\overrightarrow b .\,\overrightarrow b } \right)\left( {\overrightarrow b \times \overrightarrow a } \right) - \left( {\overrightarrow a .\,\overrightarrow b } \right)\left( {\overrightarrow b \times \overrightarrow a } \right)$$

$$ = \left( {\overrightarrow b \times \overrightarrow a } \right)\left( {\overrightarrow b .\,\overrightarrow b - \overrightarrow a .\,\overrightarrow a } \right)$$

$$ = \left( {5\overrightarrow b \times \overrightarrow a } \right)\left( {5 + {\lambda ^2} - 13 - {\lambda ^2}} \right)$$

$$ = 8\left( {\overrightarrow a \times \overrightarrow b } \right)$$

$$\overrightarrow a \times \overrightarrow b = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr \lambda & 2 & { - 3} \cr 1 & { - \lambda } & 2 \cr } } \right|$$

$$ = \widehat i(4 - 3\lambda ) - \widehat j(2\lambda + 3) + \widehat k( - {\lambda ^2} - 2)$$

$$ \Rightarrow \lambda = 1$$

$${\left| {\overrightarrow a \times \left( {\overrightarrow a - \overrightarrow b } \right) + \overrightarrow b \times \left( {\overrightarrow a - \overrightarrow b } \right)} \right|^2}$$

$$ = {\left| {2\left( {\overrightarrow a \times \overrightarrow b } \right)} \right|^2} = 4\,.\,35 = 140$$

Vector Algebra

Continue this paper

Similar Vector Algebra PYQs

More practice from this question