The angle ϕ between a line and a plane is measured as the complement of the angle between the line’s direction vector d and the plane’s normal vector n.

Since the normal forms an angle of 90° with the plane, we can construct a right angled triangle using the intersection of the normal, the line, and the plane:

If θ is the angle between d and n, then ϕ=180°−90∘−θ=90°−θ

In practice, you can compute θ using cosθ=∣d∣∣n∣d⋅n​.

Example

Consider the line r=⎝⎛​000​⎠⎞​+λ⎝⎛​231​⎠⎞​ and the plane with normal n=⎝⎛​314​⎠⎞​. The direction vector is d=⎝⎛​231​⎠⎞​.

The angle θ between d and n satisfies

So θ=47.05°, thus ϕ=90°−47.05°=43.0°