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How to Determine the Angle Between Two Lines

When you want to know the angle between two lines, you can find the angle between their directional vectors. If you have a line l along the vector rl = (a,b,c) and a line m along the vector rm = (d,f,g), you can find the angle α between the two lines this way:

Formula

The Angle Between Two Lines

cos α = rl rm |rl| |rm| ,α [0°, 180°]

Note! If α > 90°, the real angle between the lines is β = 180° α. This is because the angle between two lines always is 90°.

Example 1

You have a line l along the vector rl = (2, 3, 4) and a line m along the vector rm = (1,2, 1). The angle between them is cos α = (2, 3, 4) (1,2, 1) | (2, 3, 4)| | (1,2, 1)| = 2 1 + 3 (2) + 4 1 22 + 32 + 42 12 + (2 ) 2 + 12 = 2 6 + 4 29 6 = 0, α = cos 1 (0) = π 2 = 90°.

The lines are perpendicular!

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