The sine of the angle between the vectors `a=3hati+hatj+2hatk and b=2hati-2hatj+4hatk` is

Options

A. `sqrt((2)/(7))`
B. `(2)/(sqrt7)`
C. `(sqrt2)/(7)`
D. None of these

Answer

B

Explanation

Given that, `a=3 hat(i)+hat(j)+2hat(k)`
and `b=2 hat(i)-2hat(j)+4hat(k)`
Let `theta` be the angle between them.
`:. cos theta=(a.b)/(|a||b|)=((3hat(i)+hat(j)+2hat(k)))/sqrt(9+1+4)((2hat(i)-2hat(j)+4hat(k)))/sqrt(4+4+16)`
`implies cos theta=(6+(-2)+8)/(sqrt(14)sqrt(24))=12/(4sqrt(21))`
`:. sin theta=sqrt(1-cos^(2) theta)`
`=sqrt(1-144/336)=sqrt(192/336)`
`implies sin theta =2/sqrt(7)`