Let $\mathbf{a}=\hat{i}+x \hat{j}+\hat{k}, \mathbf{b}=\hat{i}+\hat{j}+\hat{k}$ and…
Let $\mathbf{a}=\hat{i}+x \hat{j}+\hat{k}, \mathbf{b}=\hat{i}+\hat{j}+\hat{k}$ and $|\mathbf{a}+\mathbf{b}|=|\mathbf{a}|+|\mathbf{b}|$, then
$x=1$
$x=-1$
$x=0$
No such real $x$ exist
Solution
$|\mathbf{a}+\mathbf{b}|=|\mathbf{a}|+|\mathbf{b}| \Rightarrow \mathbf{a}, \mathbf{b}$ are in same direction.
$
\frac{1}{1}=\frac{x}{1}=\frac{1}{1} \Rightarrow x=1
$