Question
Let $$\alpha ,\,\beta ,\,\gamma $$ be distinct real numbers. The points with position
vectors of $$\alpha \hat i + \,\beta \hat j + \,\gamma \hat k,\,\,\beta \hat i + \,\gamma \hat j + \alpha \hat k,\,\,\gamma \hat i + \alpha \hat j + \,\beta \hat k$$
A.
are collinear
B.
form an equilateral triangle
C.
form a scalene triangle
D.
form a right angled triangle
Answer :
form an equilateral triangle
Solution :
Let the given position vectors be of point $$A,\,B$$ and $$C$$ respectively, then
$$\eqalign{
& \left| {\overrightarrow {AB} } \right| = \sqrt {{{\left( {\beta - \alpha } \right)}^2} + {{\left( {\,\gamma - \,\beta } \right)}^2} + {{\left( {\alpha - \,\gamma } \right)}^2}} \cr
& \left| {\overrightarrow {BC} } \right| = \sqrt {{{\left( {\gamma - \beta } \right)}^2} + {{\left( {\alpha - \,\gamma } \right)}^2} + {{\left( {\,\alpha - \,\beta } \right)}^2}} \cr
& \left| {\overrightarrow {CA} } \right| = \sqrt {{{\left( {\alpha - \,\gamma } \right)}^2} + {{\left( {\beta - \alpha } \right)}^2} + {{\left( {\,\gamma - \,\beta } \right)}^2}} \cr
& \left| {\overrightarrow {AB} } \right| = \left| {\overrightarrow {BC} } \right| = \left| {\overrightarrow {CA} } \right| \cr} $$
$$ \Rightarrow \,\,\Delta ABC$$ is an equilateral $$\Delta .$$