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

MCQ

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

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
(b) Let $P,\,Q$ and $R$ be points having position vectors $\alpha \,i + \beta \,j + \gamma \,k,$ $\beta \,i + \gamma \,j + \alpha \,k$ and $\gamma \,i + \alpha j + \beta k$ respectively.

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

Hence $\Delta PQR$ is an equilateral triangle.

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