The sine of the angle between the line `(x-2)/(3) = (y-3)/(4) = (z-4)/(5)` and the plane `2x-2y+z=5` is

Options

A. `10/(6sqrt(5))`
B. `4/(5sqrt(2))`
C. `(sqrt(2))/10`
D. `(2sqrt(3))/5`

Answer

C

Explanation

The line is parallel to the vector `vecb=3hati+4hatj+5hatk` and the plane is normal to the vector `vecn=2hati-2hatj+hatk`.
If `theta` is the angle between the line and the plane. Then
`sin theta=(vecb.vecn)/(|vecb||vecn|)impliessin theta=(6-8+5)/(sqrt(50)sqrt(9))=1/(5sqrt(2))=(sqrt(2))/10`