The value of 'a' for which the points A, B,C with position vectors ` 2hati - hatj + hatk , hati + 2hatk, hati -3hatj - 5hatk and a hati -3hatj +hatk` respectively are the vertices of a right angled triangle with ` C = pi//2`, are

Options

A. `-2 and -1`
B. `-2 and 1`
C. `2 and -1`
D. 2 and 1

Answer

D

Explanation

Given, position vectors of A,B, C are `2hati - hatj + hatk`, `hati - 3hatj - 5hatk` and `ahati - 3hatj + hatk`, respectively.
Now, `AC = (ahati- 3hatj + hatk) - (2hati - hatj + hatk) = (a-2) hati - 2hatj`
and `BC = (ahati - 3hatj + hatk) - (hati - 3hatj - 5hatk) = (a-1)hati + 6hatk`
Since, the `DeltaABC` is right angled at C, therefore
`AC.BC = 0`
`rArr {(a-2)hati- 2hatj}.{(a-1)hati + 6hatk} = 0`
`rArr (a-2)(a-1)= 0`
`:. a = 1` and `2`