Question

The perimeter of the triangle whose vertices have the position vectors $(\mathbf{i} + \mathbf{j} + \mathbf{k}),(5\mathbf{i} + 3\mathbf{j} - 3\mathbf{k})$ and $(2\mathbf{i} + 5\mathbf{j} + 9\mathbf{k}),$ is given by

• A. $15 + \sqrt{157}$
• B. $15 - \sqrt{157}$
• C. $\sqrt{15} - \sqrt{157}$
• D. $\sqrt{15} + \sqrt{157}$

Answer 1

Answer: (A)

$15 + \sqrt{157}$

Answer 2

$\mathbf{a} = 4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k} \Rightarrow |\mathbf{a}| = \sqrt{16 + 16 + 4} = 6$

$\mathbf{b} = - 3\mathbf{i} + 2\mathbf{j} + 12\mathbf{k} \Rightarrow |\mathbf{b}| = \sqrt{144 + 4 + 9} = \sqrt{157}$

$\mathbf{c} = - \mathbf{i} - 4\mathbf{j} - 8\mathbf{k} \Rightarrow |\mathbf{c}| = \sqrt{64 + 16 + 1} = 9$

Hence perimeter is $15 + \sqrt{157}.$