Let , , be distinct real numbers. The points with position vector… - Vidyadip

MCQ

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

A
Are collinear
✓
Form an equilateral triangle
C
Form a scalene triangle
D
Form a right angled triangle

✓

Answer

Correct option: B.

Form an equilateral triangle

(B) Let $P , Q$ and R be points having position vectors $\alpha \hat{ i }+\beta \hat{ j }+\gamma \hat{ k }, \beta \hat{ i }+\gamma \hat{ j }+\alpha \hat{ k }$ and $\gamma \hat{ i }+\alpha \hat{ j }+\beta \hat{ k }$
Then, $|\overline{ PQ }|=|\overline{ QR }|=|\overline{ RP }|$
$=\sqrt{(\alpha-\beta)^2+(\beta-\gamma)^2+(\gamma-\alpha)^2}$
Hence, $\triangle PQR$ is an equilateral triangle.

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