1.

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

Answer»

`-2 and -1`
`-2 and 1`
`2 and -1`
2 and 1

Solution :Given, POSITIONVECTORS 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 rightangledat 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`


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