How do I find the angle between two lines using their direction vectors?

Represent each line by its direction vector, taken from the direction ratios. If line 1 has direction vector a and line 2 has direction vector b, the angle θ between the lines satisfies cos θ = |a·b| divided by (|a||b|). The absolute value in the numerator ensures you get the acute angle between the lines, which is the convention. Compute a·b as the sum of the products of corresponding direction ratios, and the magnitudes as the square roots of the sums of squares. For example if a has ratios (1, 2, 2) and b has (2, 2, 1), then a·b = 2 + 4 + 2 = 8, each magnitude is 3, so cos θ = 8/9 and θ = arccos(8/9). If a·b = 0 the lines are perpendicular.