Projection keeps the part of one vector that points along another chosen nonzero vector; the remainder is perpendicular.
Use it to separate a component along a direction from the leftover orthogonal component.
The relationship
The defining relationship
projᵥ(u)=(u·v/v·v)v
See the structure
What the calculation is doing
input vector× Aoutput vectorprojᵥ(u)=(u·v/v·v)v
Worked interpretation
Read the result in context
projᵥ(u)=(u·v/v·v)v
Interpret with care
Important boundary
The direction vector v must be nonzero, and projection depends on the chosen dot product.
Use the Matrix Calculator to experiment with small exact matrix operations before applying a more specialized route.
Quick guide
How to use this calculator
Enter vector a to project.
Enter a nonzero direction vector b with the same dimension.
Read the exact vector projection, its coefficient, and the scalar projection.
Calculation method
Scale the direction vector
The vector projection of a onto b is ((a·b)/(b·b))b. The scalar projection is (a·b)/‖b‖ and may require a labelled square-root approximation. Projection onto the zero vector is undefined.
Worked example
Project onto the x-axis
Project a = ⟨3,4⟩ onto b = ⟨1,0⟩.
proj_b(a) = 3⟨1,0⟩ = ⟨3,0⟩
Supported inputs
Precision and limits
Vector size
General vectors may contain 2–8 components; three-dimensional products require exactly three.
Entry precision
Each signed decimal may contain up to 30 digits and 15 decimal places.
Square roots
Squared magnitudes remain exact. Irrational roots and normalized components are explicitly labelled approximations.
