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Area of Triangle with Coordinates of Vertices Given

Let α, β an...

Question

Let $α, β$ and $γ$ be distinct and real numbers. The points with position vectors $α i + β ˆ j + γ ˆ k, β ˆ i + γ ˆ j + α ˆ k$ and $γ ˆ i + α ˆ j + β ˆ k$

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 given position vectors be of points $A, B$ and $C$ , respectively, then
$| \to A B | = \sqrt{(β - α)^{2} + (γ - β)^{2} + (α - γ)^{2}}$
$| \to B C | = \sqrt{(γ - β)^{2} + (α - γ)^{2} + (α - β)^{2}}$
$| \to C A | = \sqrt{(α - γ)^{2} + (β - α)^{2} + (γ - β)^{2}}$
$∵ | \to A B | = | \to B C | = | \to C A |$
Hence, $Δ A B C$ is an equilateral triangle.

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Area of Triangle with Coordinates of Vertices Given

Standard XII Mathematics

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