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Let vec a=2 hat i- hat j+ hat k , vec b...

Let ` vec a=2 hat i- hat j+ hat k , vec b= hat i+2 hat j- hat k` and ` vec c= hat i+ hat j-2 hat k` be three vectors. A vector in the plane of ` vec b` and `vec c` , whose projection on ` vec a` is of magnitude `sqrt(2//3)` , is

A

`2hat(i) + 3hat(j)-3hat(k)`

B

`2hat(i)+3hat(j)+3hat(k)`

C

`-2hat(i)-hat(j)+5hat(k)`

D

`2hat(i)+hat(j)+5hat(k)`

Text Solution

Verified by Experts

The correct Answer is:
A, C
Given vectors are `vec(a) = 2hat(j) +hat(k) , hat(b) = hat(i) +2hat(j) - hat(k) "and " vec( c) =hat(i) + hat(j) -2 hat(k)` . Any vector `vec(r )` in the plane of `vec(b) ` and `vec(c )` is
` vec( r ) = vec(b) + t(vec( c ))= hat(i) +2hat(j) - hat(k) + t (hat(i) +hat(j) -2hat(k))`
`(1+t) hat(i) +(2 +t) hat(j) -(1+2t)hat(k)`
Since projection of `vec(r ) " on " vec(a) " is " sqrt((2)/(3))`
`:. (vec(r ). vec(a))/(|vec(a)|)= sqrt((2)/(3))`
`rArr |(2(1+t)-(2+t)-(1+2t))/(sqrt(6))| =sqrt((2)/(3))`
`rArr |-(1+t) | =2 rArr t=1 " or " -3`
On putting t=1 , -3 and Eq. (i) respectively we get
`vec( r)= 2hat(i)+ 3hat(j) -3hat(k)`
` vec(r ) = 2hat(i) - hat(j) + 5ha(k)`
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