Angle between a Line and a Plane

Let the line be $\vec r=\vec a+\lambda\vec b$ and the plane be $\vec r\cdot\vec n=d$. Two extreme relationships are worth naming first. The line is perpendicular to the plane exactly when its direction $\vec b$ is parallel to the plane's normal $\vec n$, i.e. when $\vec b=t\vec n$ for some real number $t$. The line is parallel to the plane (running flat alongside it, making no net approach toward or away from it) exactly when $\vec b$ is perpendicular to $\vec n$, i.e. $\vec b\cdot\vec n=0$.

For every other case, the angle between the line and the plane is defined as the complement of the acute angle between the line and the normal to the plane. Because it is built as a complementary angle to an acute angle, the angle between a line and a plane can never be obtuse — it always lies strictly between $0^\circ$ and $90^\circ$. If $\theta$ is the required angle, the acute angle between $\vec b$ and $\vec n$ is $\dfrac{\pi}{2}-\theta$, so

Worked example. Find the angle between the line $\vec r=(\hat i+2\hat j+\hat k)+\lambda(\hat i+\hat j+\hat k)$ and the plane $\vec r\cdot(2\hat i-\hat j+\hat k)=8$.

Here $\vec b=\hat i+\hat j+\hat k$ and $\vec n=2\hat i-\hat j+\hat k$.

$\vec b\cdot\vec n=(1)(2)+(1)(-1)+(1)(1)=2-1+1=2$. …