If `hat(i), hat(j)` and `hat(k)` represent unit vectors along the x, y and z axes respectively, then the angle `theta` between the vectors `(hat(i) + hat(j) + hat(k))` and `(hat(i) + hat(j))` is equal to :

Options

A. `sin^(-1) ((1)/(sqrt3))`
B. `sin^(-1) (sqrt((2)/(3)))`
C. `cos^(-1) ((1)/(sqrt3))`
D. `90^(@)`

Answer

A

Explanation

Since `vec(A).vec(B)= AB cos theta`
`rArr cos theta= (vec(A).vec(B))/(AB)= ((hat(i) + hat(j) + hat(k))(hat(i) + hat(j)))/(sqrt(1^(2) + 1^(2) +1^(2)) sqrt(1^(2) + 1^(2)))`
`rArr cos theta= (2)/(sqrt6) = sqrt((2)/(3))`
`rArr sin theta= (1)/(sqrt3)`
`rArr theta= sin^(-1) ((1)/(sqrt3))`