Question

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

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

Answer 1

The correct option is B Form an equilateral triangle

Let the given position vectors be of points $A,B$ and $C$, respectively, then

$|\overrightarrow{AB}|=\sqrt{(\beta -\alpha {)}^2+(\gamma -\beta {)}^2+(\alpha -\gamma {)}^2}$
$|\overrightarrow{BC}|=\sqrt{(\gamma -\beta {)}^2+(\alpha -\gamma {)}^2+(\alpha -\beta {)}^2}$
$|\overrightarrow{CA}|=\sqrt{(\alpha -\gamma {)}^2+(\beta -\alpha {)}^2+(\gamma -\beta {)}^2}$
$\because |\overrightarrow{AB}|=|\overrightarrow{BC}|=|\overrightarrow{CA}|$

Hence, $\Delta ABC$ is an equilateral triangle.