Question

Let $\alpha,\beta,\gamma$ be distinct real numbers. The points with position vectors $\alpha\mathbf{i} + \beta\mathbf{j} + \gamma\mathbf{k},\beta\mathbf{i} + \gamma\mathbf{j} + \alpha\mathbf{k},\gamma\mathbf{i} + \alpha\mathbf{j} + \beta\mathbf{k}$

• A. Are collinear
• B. Form an equilateral triangle
• C. Form a scalene triangle
• D. Form a right angled triangle

Answer 1

Answer: (B)

Form an equilateral triangle

Answer 2

Let $P,Q$ and $R$ be points having position vectors $\alpha\mathbf{i} + \beta\mathbf{j} + \gamma\mathbf{k},$ $\beta\mathbf{i} + \gamma\mathbf{j} + \alpha\mathbf{k}$ and $\gamma\mathbf{i} + \alpha j + \beta\mathbf{k}$ respectively.

Then,$|\overset{\rightarrow}{PQ}| = |\overset{\rightarrow}{QR}| = |\overset{\rightarrow}{RP}| = \sqrt{(\alpha - \beta)^{2} + (\beta - \gamma)^{2} + (\gamma - \alpha)^{2}}$

Hence $\Delta PQR$ is an equilateral triangle.