Explanation

(b)
Let $\overrightarrow{r} \times \overrightarrow{a} = \overrightarrow{b} \times \overrightarrow{a}$
$\Rightarrow \left( \overrightarrow{r} - \overrightarrow{b} \right) \times \overrightarrow{a} = \overrightarrow{0}\ \Rightarrow \ \overrightarrow{r} = \overrightarrow{b} + t\overrightarrow{a}\$
Similarly, other line $\overrightarrow{r} = \overrightarrow{a} + k\overrightarrow{b}$, where $t$ and $k$ are scalars
Now $\overrightarrow{a} + k\overrightarrow{b} = \overrightarrow{b} + t\overrightarrow{a}$
$\Rightarrow \ t = 1,k = 1$
(equating the coefficients of $\overrightarrow{a}$ and $\overrightarrow{b})$
$\therefore\ \overrightarrow{r} = \overrightarrow{a} + \overrightarrow{b} = \widehat{i} + \widehat{j} + 2\widehat{i} - \widehat{k} = 3\widehat{i} + \widehat{j} - \widehat{k}$
i.e., $(3,\ 1,\ - 1)$