The Angle between two Lines

Problem: Find the angle between the line parallel to the vector [1, -1, 1] and the line parallel to the vector [-6, -4, 0]
A diagram of this is shown on the right.		is the vector to which one line is parallel. is the vector to which the other line is parallel. θ is the angle between the two lines.
The scalar (dot) product of two vectors is defined as; the product of their magnetude and the cosine of their included angle; The scalar (dot) product of and is therefore; . = \| \| × \| \|cosθ
Using . = \| \| × \| \|cosθ the required angle can be found.	The scalar (dot) product . = [1, -1, 1] . [-6, -4, 0] = 1 × -6 + -1 × -4 + 1 × 0 = -2 The product \| \| × \| \| = × = 2 We can now calculate the cosine of the angle between the two vectors using; . = \| \| × \| \|cosθ -2 = 2cosθ θ = = 1.73 radians The required angle is 1.73 radians.

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