How to Use the Vector Projection Calculator

Enter vector A and vector B. The projection of A onto B — both as a vector and as a scalar length — comes out below.

scalar projection = (A · B) ÷ |B|
vector projection = ((A · B) ÷ |B|²) × B

Projecting A onto B answers "how much of A points in B's direction" — picture shining a light straight down onto B and looking at the shadow A casts across it. The scalar version is just that shadow's length; the vector version is the same shadow expressed as an actual vector lying along B. When A and B are perpendicular, the shadow disappears entirely and the projection is zero, matching the dot product going to zero at the same angle.