Let $\overrightarrow b = \widehat i + \widehat j + \lambda \widehat k$, $\lambda$ $\in$ R. If $\overrightarrow a$ is a vector such that $$\overrightarrow a \times \overrightarrow b = 13\widehat i - \widehat j - 4\widehat k$$ and $\overrightarrow a \,.\,\overrightarrow b + 21 = 0$, then $$\left( {\overrightarrow b - \overrightarrow a } \right).\,\left( {\widehat k - \widehat j} \right) + \left( {\overrightarrow b + \overrightarrow a } \right).\,\left( {\widehat i - \widehat k} \right)$$ is equal to _____________.

<p>Let $\overrightarrow a = x\widehat i = y\widehat j + z\widehat k$</p> <p>So, $$\left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr x & y & z \cr 1 & 1 & \lambda \cr } } \right| = \widehat i(\lambda y - z) + \widehat j(z - \lambda x) + \widehat k(x - y)$$</p> <p>$\Rightarrow \lambda y - z = 13,\,z - \lambda x = - 1,\,x - y = - 4$</p> <p>and $x + y + \lambda z = - 21$</p> <p>$\Rightarrow$ Clearly, $\lambda = 3$, $x = - 2$, $y = 2$ and $z = - 7$</p> <p>So, $$\overrightarrow b - \overrightarrow a = 3\widehat i - \widehat j + 10\widehat k$$</p> <p>and $$\overrightarrow b + \overrightarrow a = - \widehat i + 3\widehat j - 4\widehat k$$</p> <p>$$ \Rightarrow \left( {\overrightarrow b - \overrightarrow a } \right)\,.\,\left( {\widehat k - \widehat j} \right) + \left( {\overrightarrow b + \overrightarrow a } \right)\,.\,\left( {\widehat i - \widehat k} \right) = 11 + 3 = 14$$</p>