Find the area of the parallelogram whose one side and a diagonal are represented by coinitial vectors $\hat{i} - \hat{j} + \hat{k}$ and $4\hat{i} + 5\hat{k}$ respectively.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → $\sqrt{42}$ sq. units ##
Let $\vec{a} = \hat{i} - \hat{j} + \hat{k}$
$\vec{d} = 4\hat{i} + 5\hat{k}$
$∵\vec{a} + \vec{b} = \vec{d}$
$∴\vec{b} = \vec{d} - \vec{a} = 3\hat{i} + \hat{j} + 4\hat{k}$
$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -1 & 1 \\ 3 & 1 & 4 \end{vmatrix} = -5\hat{i} - 1\hat{j} + 4\hat{k}$
Area of parallelogram $= |\vec{a} \times \vec{b}| = \sqrt{25 + 1 + 16} = \sqrt{42}$ sq. units.