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Distance between Two Points on the Same Coordinate Axes

Let α ,β ,γ...

Question

Let $α, β, γ$ be distinct real numbers. The point with positions vectors $α^i + β^j + γ^k$ , $β^i + γ^j + α^k, γ^i + α^j + β^k$

A

Are collinear

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B

Form an equilateral triangle

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C

Form a scalene triangle

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D

Form a right angled triangle

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Solution

The correct option is B Form an equilateral triangle
Let the points be denoted by $A, B$ and $C$ . Then
$| A B | = \sqrt{{(β - α)}^{2} + {(γ - β)}^{2} + {(α - γ)}^{2}} | B C | = \sqrt{{(γ - β)}^{2} + {(α - γ)}^{2} + {(α - β)}^{2}} | C A | = \sqrt{{(α - γ)}^{2} + {(β - α)}^{2} + {(γ - β)}^{2}}$
Since $A B = B C = C A$ , we get $A B C$ is an equilateral triangle.

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Distance between Two Points on the Same Coordinate Axes

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