Find the angle between the line `(x-1)/3=(y-1)/2=(z-1)/4` and the plane `2x+y-3z+4=0.`

Explanation

The angle between a line and a plane is complement of the angle between the line and the normal of the plane, i.e., 3, 2, 4 and normal 2, 1, -3. Therefore,
`" "costheta=(6+2-12)/(sqrt(29)*sqrt(14))=-(4)/(sqrt(406))`
`" "theta=cos^(-1) (-4//sqrt(406))`
`" "phi=90^(@)-theta`
`" " = 90^(@)- cos^(-1) * (-4//sqrt(406))`
`" " = sin ^(-1) (-4//sqrt(406))`