The vectors $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$, $\vec{b} = \hat{i} - 3\hat{j} - 5\hat{k}$ and $\vec{c} = -3\hat{i} + 4\hat{j} + 4\hat{k}$ represent the sides of:

a right-angled triangle

The correct answer is Option (4) → a right-angled triangle ##

$\vec{a} = 2\hat{i} - \hat{j} + \hat{k},$

$\vec{b} = \hat{i} - 3\hat{j} - 5\hat{k},$

$\vec{c} = -3\hat{i} + 4\hat{j} + 4\hat{k},$

When two vectors are perpendicular to each other:

$\vec{A} \cdot \vec{B} = 0$

So, $\vec{a} \cdot \vec{b} = (2\hat{i} - \hat{j} + \hat{k}) \cdot (\hat{i} - 3\hat{j} - 5\hat{k})$

$= [2 + 3 - 5]$

$\vec{a} \cdot \vec{b} = 0$

So, a right-angled triangle is formed.