Let `alpha, beta and gamma` be distinct real numbers. The points whose position vector's are `alpha hat i + beta hat j+gamma hat k; beta hat i+gamma hat j+alpha hat k and gamma hat i+alpha hat j+beta hat k` a. are collinear. b. forms an equilateral triangle. c. forms a scalene triangle. d. forms a right angled triangle.

Options

A. are collinear
B. form an equilateral triangle
C. form a scalene triangle
D. form a right-angled triangle

Answer

B

Explanation

Let the given position vectors be of points A, B and C, respectively. Then
`|vec(AB)| = sqrt((beta -alpha)^(2)+ (gamma-beta)^(2) + (alpha -gamma)^(2))`
`|vec(BC)| = sqrt((gamma-beta)^(2)+ (alpha -gamma)^(2) + (alpha -beta)^(2))`
`|vec(CA)| = sqrt((alpha-gamma)^(2)+ (beta-alpha)^(2)+ (gamma-beta)^(2))`
`because |vec(AB)| = |vec(BC)| = |vec(CA)|`
Hence, `DeltaABC` is an equilateral triangle.