Vector Projection Calculator
Project one vector onto another and compute the scalar and vector projections.
Inputs
proj_B(A) (vector projection)
[3, 0]
Scalar projection
3.000000
Orthogonal component
[0, 4]
Step by step
A
= [3, 4]
B
= [1, 0]
A·B
= 3
||B||²
= 1
proj_B(A) = (A·B/||B||²) × B
= [3, 0]
Orthogonal remainder A − proj
= [0, 4]
How it works
The vector projection of A onto B is the component of A in the direction of B: proj_B(A) = (A·B / ||B||²) × B. The scalar projection is A·B / ||B|| (the signed length of the shadow). The orthogonal component A − proj_B(A) is perpendicular to B.
Formulas
Vector Projection
proj_B(A) = (A·B / ||B||²) × B
- A
- Vector to project
- B
- Direction vector
Scalar Projection
comp_B(A) = A·B / ||B||
- A
- Vector to project
- B
- Direction vector
Frequently Asked Questions
What is the difference between scalar and vector projection?
Scalar projection is the signed magnitude (a number). Vector projection is the actual vector in the direction of B whose length equals the scalar projection.
What is the orthogonal component?
It is A minus the projection: the part of A perpendicular to B. Together, the projection and orthogonal component reconstruct A.