Question

Let `alpha, beta, gamma` be distinct real numbers. The points with position vectors `alpha hati + beta hatj +gamma hat k , beta hati + gamma hatj +alpha hat k , gamma hati +alpha hatj + beta hatk`
A. are collinera
B. form an equilateral triangle
C. form a scalene triangle
D. form a right angled triangle

Answer 1

Let given point be A, B and C with positive vectors `alphahati+betahatj+gammahatk,betahati+gammahatj+alphahatk and gamma hati+alphahatj+betahatk`
As `alpha,beta and gamma` are distinct real numbers, therefore ABC form a triangle.
Clearly, `AB=OB-OA=(betahati+gammahatj+alphahatk)-(alphahati+betahatj+gammahatk)`
`=(beta-alpha)hati+(gamma-beta)hatj+(alpha-gamma)hatk`

Now, `|AB|=sqrt((beta-alpha)^(2)+(gamma-beta)^(2)+(alpha-gamma)^(2))`
Similarly, `BC=CA=sqrt((beta-alpha)^(2)+(gamma-beta)^(2)+(alpha-gamma)^(2))`
`thereforeDeltaABC` is an equilateral triangle.