# How Do You Find the Angle Between a Line and a Plane?

When you want to find the angle between a line and a plane, you do similar things to when you find the angle between two lines. Instead of looking at the directional vectors of two lines, you look at the directional vector of the line ${\stackrel{\to }{r}}_{l}$ and the normal vector to the plane ${\stackrel{\to }{n}}_{\beta }$. It’s important to remember this, though:

• If $\gamma <90\text{°}$, the angle between the line and the plane is $\alpha =90\text{°}-\gamma$.

• If $\gamma >90\text{°}$, the angle between the line and the plane is $\alpha =\gamma -90\text{°}$.

The angle between a line and a plane is always $\le 90\text{°}$.

Example 1

The line $l$ goes along the vector ${\stackrel{\to }{r}}_{l}=\phantom{\rule{-0.17em}{0ex}}\left(2,3,4\right)$, and the plane $\beta$ has a normal vector ${\stackrel{\to }{n}}_{\beta }=\phantom{\rule{-0.17em}{0ex}}\left(1,1,1\right)$. The angle between the two vectors is $\begin{array}{llll}\hfill \mathrm{cos}\gamma & =\frac{\phantom{\rule{-0.17em}{0ex}}\left(2,3,4\right)\cdot \phantom{\rule{-0.17em}{0ex}}\left(1,1,1\right)}{\phantom{\rule{-0.17em}{0ex}}|\phantom{\rule{-0.17em}{0ex}}\left(2,3,4\right)|\cdot \phantom{\rule{-0.17em}{0ex}}|\phantom{\rule{-0.17em}{0ex}}\left(1,1,1\right)|}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{2\cdot 1+3\cdot 1+4\cdot 1}{\sqrt{{2}^{2}+{3}^{2}+{4}^{2}}\cdot \sqrt{{1}^{2}+{1}^{2}+{1}^{2}}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{2+3+4}{\sqrt{29}\cdot \sqrt{3}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{9}{\sqrt{87}},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \gamma & ={\mathrm{cos}}^{-1}\phantom{\rule{-0.17em}{0ex}}\left(\frac{9}{\sqrt{87}}\right)\approx 15.23\text{°}.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Because $15.23\text{°}<90\text{°}$, you do this:

 $\alpha =90\text{°}-15.23\text{°}=74.77\text{°}.$

That gives you the angle between the line and the plane to be $\alpha =74.77\text{°}$.