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.

Try:
Answerproj_b(a) = (3, 0)
  1. Vectorsa = (3, 4), b = (1, 0)
  2. Dot producta · b = 3
  3. Magnitude squared|b|² = 1
  4. Scalar projectioncomp_b(a) = (a · b)/|b| = 3
  5. Coefficient(a · b)/|b|² = 3
  6. Projection onto bproj_b(a) = (3, 0)
  7. Perpendicular componenta − proj_b(a) = (0, 4)
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Worked examples

Project (3, 4) onto (1, 0) Find the vector projection and the perpendicular component.

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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