If `vec(A)=2hat(i)-2hat(j)-hat(k)` and `vec(B)=hat(i)+hat(j)`, then:
(i) Find angle between `vec(A)` and `vec(B)`
(ii) Find the projection of resultant vector of `vec(A)` and `vec(B)` on x-axis.
(iii) Find a vector which is, if added to `vec(A)`, gives a unit vector along y-axis.

Answer

(i) ` theta=90^(@)` (ii) `3`, (iii) `-2hat(i)+3hat(j)+hat(k)`

Explanation

(i) Let the angle between `vec(A)` and `vec(B)` is `theta` then
`cos theta=(vec(A).vec(B))/(AB)=((2hat(i)-2hat(j)-hat(k)).(hat(i)+hat(j)))/(|2hat(i)+2hat(j)-hat(k)|.|hat(i)+hat(j)|)=0/(3sqrt(2))=0`
`rArr theta=90^(@)`
(ii) Resultant
`(vec(R))=vec(A)+vec(B)=(2hat(i)-2hat(j)-hat(k))+(hat(i)+hat(j))=3hat(i)=hat(j)-hat(k)`
Projection of resulatnt on x-axis `=3`
(iii) Required vector
`=hat(j)-vec(A)=hat(j)-(2hat(i)-2hat(j)-hat(k))=-2hat(i)+3hat(j)+hat(k)`