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Let overset(to)(a) = hat(i) - hat(j) ,...

Let `overset(to)(a) = hat(i) - hat(j) , vecb=-hat(j) - hat(k) , overset(to)( c) =- hat(i) - hat(k) .` If `overset(to)(d)` is a unit vector such that `overset(to)(a).vec(d) =0= [ vec(b) vec(c ) vecd]` then `overset(to)(d)` equals

A

`+- . (hat(i)+hat(j)-2hat(k))/(sqrt(6))`

B

`+- . (hat(i) + hat(j) - hat(k))/(sqrt(3))`

C

` +- . (hat(i) + hat(j) + hat(k))/(sqrt(3))`

D

` +- hat(k)`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `vec(d) = xhat(i) +yhat(j) +zhat(k)`
where `x^(2) + y^(2) + z^(2) =1`
Since `vec(a) ". " vec(d) =0`
` rArr x-y =0rArr x=y`
Also `[vec(b) vec(c ) vec(d)] =0`
`rArr |{:(0,,1,,-1),(-1,,0,,1),(x,,y,,z):}|=0 rArr x+y+ z=0`
`rArr 2x+ z=0 `
From Eqs (i) , (ii) and (iii)
`x^(2) +x^(2) +4x^(2) = 1 rArr x = +- (1)/(sqrt(6))`
`:. vec(d) = +- (1)/(sqrt(6)) (hat(i) +hat(j) - 2hat(k))`
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