Angle Between Two 2D Vectors (Components)

Divide the dot product by both magnitudes and what is left is a pure cosine, so the angle between any two vectors falls out of their coordinates with no drawing required. Take a = (1, 0) and b = (1, 1): the dot product is 1, the magnitudes are 1 and √2, and arccos(1/√2) = 45°, which is obviously right. Take a = (3, 4) and b = (4, 3): the dot product is 24, both magnitudes are 5, and arccos(24/25) = arccos(0.96) ≈ 16.26° — a result no sketch would give you to two decimals.

The Cauchy–Schwarz inequality guarantees the fraction never leaves the range −1 to 1, so an angle always exists. Floating-point arithmetic does not care: for two nearly parallel vectors the computed ratio can come out as 1.0000000000000002, and arccos of that is NaN. This is one of the most reported bugs in graphics and robotics code, and the fix is a one-line clamp before the arccos — which formula.expert applies. Note also that the answer runs from 0° to 180° and carries no sense of rotation; if you need to know whether b sits clockwise or counterclockwise from a, use the cross product's sign instead.