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Angle between lines
The angle between two lines is simply the angle between their direction vectors.
For any two lines r1=a1+λb1 and r2=a2+μb2, the angle θ between r1 and r2 can be found via the formula
which is just the equation of the scalar product.
Angle between line and plane
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.
Angle between 2 planes
The angle θ between two planes can be found by computing the angle between the two plane normals,
Acute vs Obtuse Angles
If the scalar product of two vectors is negative, then
and thus θ must be an obtuse angle: 90°<θ≤180°.
But since vectors always form an acute AND an obtuse angle, we can find the acute angle by subtracting
whenever θ>90°.
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