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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) - 14 = 0`

C

`vec(c) .hat(k) + 14 = 0 `

D

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

Text Solution

Verified by Experts

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