Finding Angle Between Vectors

Calculus 2 · Axiom Academy

EXAMPLE Finding Angle Between Vectors Compute the angle between two vectors using the dot product and their magnitudes. Find the angle between the vectors and . The angle arc is drawn at the vectors' true directions — it measures the θ ≈ 47.7° we compute below. Nicely done — you found the angle between two vectors end to end. Here's the recipe: Cosine formula: the angle satisfies . Solve for the angle: , which here gives . Range of arccos: the result is always between 0 and radians ( and ). The same recipe works in any dimension — just add one product per coordinate. You'll reuse it constantly for projections, work, and testing whether vectors are perpendicular.

This is the written version of the interactive lesson above. See the full Calculus 2 course.