Dot Product and Angle Between Vectors

Quick answer: For nonzero vectors a and b, a·b = ‖a‖‖b‖cos θ. Thus θ = arccos((a·b)/(‖a‖‖b‖)), with θ between 0° and 180°. A zero dot product means perpendicular nonzero vectors.

Worked example

Find the angle between a = (1,0) and b = (1,√3).

1. Dot product = 1×1 + 0×√3 = 1.
2. Magnitudes are 1 and √(1 + 3) = 2.
3. cos θ = 1/(1×2) = 1/2.
4. Therefore θ = 60°.

Check the result

The second vector has positive x and y components; its direction is 60° above the first.

Common mistake to avoid

Do not apply the angle formula to the zero vector because division by its magnitude is undefined.

Try it yourself

1. Find the angle between (1,2) and (2,−1). 2. Find the angle between (1,0) and (−2,0).

Answers and reasoning

1. Dot product 2 − 2 = 0, so 90°. 2. Cosine is −1, so 180°.

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