The position vector of three consecutive vertices of a parallelogram ABCD are $A(4\hat{i} + 2\hat{j} - 6\hat{k})$, $B(5\hat{i} - 3\hat{j} + \hat{k})$ and $C(12\hat{i} + 4\hat{j} + 5\hat{k})$. The position vector of D is given by

$11\hat{i} + 9\hat{j} - 2\hat{k}$

The correct answer is Option (3) → $11\hat{i} + 9\hat{j} - 2\hat{k}$ ##

In a parallelogram, the opposite sides are equal.

Thus $\vec{AB}$ is equal to $\vec{DC}$.

$\vec{AB} = \vec{DC}$

$\vec{b} - \vec{a} = \vec{c} - \vec{d}$

$\vec{d} = \vec{a} - \vec{b} + \vec{c}$

$\vec{d} = (4\hat{i} + 2\hat{j} - 6\hat{k}) - (5\hat{i} - 3\hat{j} + \hat{k}) + (12\hat{i} + 4\hat{j} + 5\hat{k})$

$\vec{d} = 11\hat{i} + 9\hat{j} - 2\hat{k}$