Let `vec(a) =hat(i)+2hat(j)+hat(k)," " vec(b) =hat(i)-hat(j)+hat(k)," "vec(c )=hat(i)+hat(j) -hat(k).`
A vector coplanar to `vec(a)` and `vec(b)` has a projections along `overset(to)(c )` of magnitude `(1)/(sqrt(3))` then the vector is

Options

A. `4hat(i) -hat(j)+4hat(k)`
B. `4hat(i)+hat(j)-4hat(k)`
C. `2hat(i)+hat(j)+hat(k)`
D. None of the these

Answer

A

Explanation

Let vector `vec(r )` be coplanar to `vec(a) " and " vec(b)`
`:. ,vec(r ) = vec(a) + tvec(b)`
`rArr vec(r ) =( hat(i) + 2hat(j) + hat(k)) +t(hat(i)- hat(j) + hat(k))`
`=hat(i) (1+t)+ hat(j) (2-t) + hat(k) (1+t)`
the projection of `vec(r ) " on " vec( c ) = (1)/(sqrt(3))`
`rArr " " (vec(r ) ". " vec(c ))/(|vec( c) |) = (1)/(sqrt(3))`
`rArr (|1.(1+t)+1. (2-t)-1.(1+t)|)|(sqrt(3)) = (1)/(sqrt(3))`
`rArr (2-t) =+- 1 rArr t=1 " or " 3`
When t=1 we have `vec(v) =2hat(i) + hat(j) + 2hat(k)`
When t=3 we have `vec(v) =4hat(i) - hat(j) + 4hat(k)`