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If angle bisector of overline(a) = 2hat(...

If angle bisector of `overline(a) = 2hat(i) + 3 hat(j) + 4 hat(k)` and `overline(b) = 4hat(i) - 2 hat(j) + 3 hat(k)` is `overline(c ) = alpha hat(i) + 2 hat(j) + beta hat(k)` then

A

`vec(c ).hat(k) + 7 = 0`

B

`vec(c ) .hat(k) - 15 = 0`

C

`vec(c ) .hat(k) + 15 = 0`

D

`vec(c ) .hat(k) - 7 = 0`

Text Solution

Verified by Experts

The correct Answer is:
b
The angle bisector of `overline(a)` and `overline(b)` is `overline(p)`
`overline(p) = lambda (hat(a) + hat(b))`
`= lambda (((2 hat(i) + 3 hat(j) + 4 hat(k)) + (4 hat(i) - 2 hat(j) + 3 hat(k)))/(sqrt(4 + 9 + 16))) = (lambda)/(sqrt(29)) [6 hat(i) + hat(j) + 7 hat(k)]`
`= (lambda)/(2 sqrt(29)) [12 hat(i) + 2 hat(j) + 14 hat(k)]`
then `overline(p) = overline(c ) implies alpha = 12 beta = 14`
Now `overline(c ). hat(k) = 14 = 0`
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