Question

The angle between line $r = a +\lambda b$ and plane $r \cdot m =d$ is given by

• A. $\cos ^{-1}\left|\frac{ b \cdot n }{| b | \cdot n \mid}\right|$
• B. $\sin ^{-1} \mid \frac{ b \times n }{ b | n |}$
• C. $\sin ^{-1}\left|\frac{ b \cdot n }{| b || n |}\right|$
• D. $\cos ^{-1}\left|\frac{ b \times n }{| b \| n |}\right|$

Answer 1

Answer: (C)

If the equation of the line is $r=a+\lambda b$ and the equation of the plane is $r \cdot n=d$. Then, the angle $\theta$ between the line and the normal to the plane is
$\cos \theta=\left|\frac{b \cdot n}{|b||n|}\right|$
and so the angle $\phi$ between the line and the plane is given by $90-\theta$, i.e.,
$\sin \left(90^{\circ}-\theta\right) =\cos \theta $
$\text { i.e., } \sin \phi =\left|\frac{b \cdot n}{|b||n|}\right| \text { or } \phi=\sin ^{-1}\left|\frac{b \cdot n}{|b||n|}\right|$