Let `alpha` be the angle which the vector `vec(V)=2hat(i)-hat(j)+2hat(k)` makes with the z-axis,. Then, what is the value of sin `alpha`?

Options

A. `1/2`
B. `1/3`
C. `sqrt(5)/3`
D. `(sqrt(5))/(9)`

Answer

C

Explanation

The given vector is `vec(V)=2hat(i)-hat(j)+2hat(k)`
and for z-axis x=0 and y=0, so the vector equation is
`vec(A)=0hat(i)+0hat(j)+hat(k)`
`cos alpha =(vec(V).vec(A))/(|vec(V)|.|vec(A)|)`
`cos alpha = (2.0+(-1).(0)+2.(1))/(sqrt(4+1+4)sqrt(0+0+1))=(2)/(3)`
Hence, `sin alpha=sqrt(1-cos^(2)alpha)`
`=sqrt(1-(4)/(9))=sqrt((5)/(9))=(sqrt(5))/(3)`