Explanation
1. Define the Points:
Let the three points be A, B, and C with position vectors:
a→=αi^+βj^+γk^
b→=βi^+γj^+αk^
c→=γi^+αj^+βk^
2. Calculate the Lengths of the Sides:
To determine the type of triangle, we find the distance between these points using the magnitude of the displacement vectors.
Side AB:
AB→=b→−a→=(β−α)i^+(γ−β)j^+(α−γ)k^
|AB→|=(β−α)2+(γ−β)2+(α−γ)2
Side BC:
BC→=c→−b→=(γ−β)i^+(α−γ)j^+(β−α)k^
|BC→|=(γ−β)2+(α−γ)2+(β−α)2
Side CA:
CA→=a→−c→=(α−γ)i^+(β−α)j^+(γ−β)k^
|CA→|=(α−γ)2+(β−α)2+(γ−β)2
3. Compare the Magnitudes:
Notice that the terms under the square root for each side length are identical, just in a different order: (α−β)2, (β−γ)2, and (γ−α)2.
Therefore:
|AB→|=|BC→|=|CA→|
4. Conclusion:
Since all three side lengths are equal and α,β,γ are distinct (ensuring the side lengths are non-zero), the points form an equilateral triangle.
Correct Option: (b) form an equilateral triangle
