Calxsolver
Linear Algebra

Vector Projection

The projection of vector a onto vector b is the component of a in the direction of b. The calculator returns the scalar projection comp_b(a), the vector projection proj_b(a) = ((a·b)/|b|²) b, and the perpendicular component a − proj_b(a).

Vector Projection

Projection of a onto b, scalar projection and the perpendicular component.

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Frequently asked questions

What is the difference between scalar projection and vector projection?

The scalar projection comp_b(a) = (a·b)/|b| is a single number giving how far a extends along b. The vector projection proj_b(a) is the actual vector in the direction of b with that length.

What if b is the zero vector?

The projection is undefined — there is no direction to project onto. The calculator reports an error in that case.

What is the perpendicular component used for?

The decomposition a = proj_b(a) + (a − proj_b(a)) splits a into a part parallel to b and a part perpendicular to b — the basis of Gram-Schmidt orthogonalisation.

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