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

How to 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 rl and the normal vector to the plane nβ. It’s important to remember this, though:

  • If γ < 90°, the angle between the line and the plane is α = 90° γ.

  • If γ > 90°, the angle between the line and the plane is α = γ 90°.

The angle between a line and a plane is always 90°.

Example 1

The line l goes along the vector rl = (2, 3, 4), and the plane β has a normal vector nβ = (1, 1, 1). The angle between the two vectors is cos γ = (2, 3, 4) (1, 1, 1) | (2, 3, 4)| | (1, 1, 1)| = 2 1 + 3 1 + 4 1 22 + 32 + 42 12 + 12 + 12 = 2 + 3 + 4 29 3 = 9 87, γ = cos 1 ( 9 87) 15.23°.

Because 15.23° < 90°, you do this:

α = 90° 15.23° = 74.77°.

That gives you the angle between the line and the plane to be α = 74.77°.

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How to Determine the Angle Between Two Lines
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