Question

The point having position vectors $2\mathbf{i} + 3\mathbf{j} + 4\mathbf{k},$ $3\mathbf{i} + 4\mathbf{j} + 2\mathbf{k},$ $4\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$ are the vertices of

• A. Right angled triangle
• B. Isosceles triangle
• C. Equilateral triangle
• D. Collinear

Answer 1

Answer: (C)

Equilateral triangle

Answer 2

Here,$\overset{\rightarrow}{OA} = 2\mathbf{i} + 3\mathbf{j} + 4\mathbf{k},$ $\overset{\rightarrow}{OB} = 3\mathbf{i} + 4\mathbf{j} + 2\mathbf{k}$

$\overset{\rightarrow}{OC} = 4\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$

So, $\overset{\rightarrow}{AB} = \mathbf{i} + \mathbf{j} - 2\mathbf{k},$ $\overset{\rightarrow}{BC} = \mathbf{i} - 2\mathbf{j} + \mathbf{k}$, $\overset{\rightarrow}{CA} = 2\mathbf{i} - \mathbf{j} - \mathbf{k}$

Clearly $|AB| = |BC| = |CA| = \sqrt{6}$

So these points are vertices of an equilateral triangle.