Let $$\overrightarrow a = x\widehat i = y\widehat j + z\widehat k$$

So, $$\left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr x & y & z \cr 1 & 1 & \lambda \cr } } \right| = \widehat i(\lambda y - z) + \widehat j(z - \lambda x) + \widehat k(x - y)$$

$$ \Rightarrow \lambda y - z = 13,\,z - \lambda x = - 1,\,x - y = - 4$$

and $$x + y + \lambda z = - 21$$

$$\Rightarrow$$ Clearly, $$\lambda = 3$$, $$x = - 2$$, $$y = 2$$ and $$z = - 7$$

So, $$\overrightarrow b - \overrightarrow a = 3\widehat i - \widehat j + 10\widehat k$$

and $$\overrightarrow b + \overrightarrow a = - \widehat i + 3\widehat j - 4\widehat k$$

$$ \Rightarrow \left( {\overrightarrow b - \overrightarrow a } \right)\,.\,\left( {\widehat k - \widehat j} \right) + \left( {\overrightarrow b + \overrightarrow a } \right)\,.\,\left( {\widehat i - \widehat k} \right) = 11 + 3 = 14$$