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Question
Let α,β,γ be distinct real numbers. The points with position vectors α^i+β^j+γ^k,β^i+γj+α^k and γ^i+α^j+β^k are....
Let α,β,γ be distinct real numbers. The points with position vectors α^i+β^j+γ^k,β^i+γj+α^k and γ^i+α^j+β^k are....
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Solution
The correct option is B Form an equilateral triangle
The 3 points given are A(α,β,γ), B(β,γ,α) and C(γ,α,β)Now, they are non-collinear as their directional unit vectors are not equivalent.
Also, AB=√(α−β)2+(β−γ)2+(γ−α)2
BC=√(β−γ)2+(γ−α)2+(α−β)2
CA=√(γ−α)2+(α−β)2+(β−γ)2
Now, as AB=BC=CA, △ABC is equilateral.
The 3 points given are A(α,β,γ), B(β,γ,α) and C(γ,α,β)
Now, they are non-collinear as their directional unit vectors are not equivalent.
Also, AB=√(α−β)2+(β−γ)2+(γ−α)2
BC=√(β−γ)2+(γ−α)2+(α−β)2
CA=√(γ−α)2+(α−β)2+(β−γ)2
Now, as AB=BC=CA, △ABC is equilateral.
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