Project vector a onto the direction of nonzero vector b. The calculation trail makes the reported vector projection easier to reproduce.
The projection of a onto b is ((a·b)/(b·b))b and lies along b.
For this page, vector projection is interpreted under the stated convention and input order.
It decomposes forces, velocities, errors, shadows, and least-squares components.
Projection onto the zero vector is undefined, and reversing b does not change the projected vector. If the Vector Projection assumptions do not fit, consider orthogonal residual.
Compare the result with the worked example's scale before relying on the reported vector projection.
Projecting (3,4,0) onto (1,1,0) gives (3.5,3.5,0). This Vector Projection example can be compared with projection factor.
Find the scalar component factor, multiply b, and check that the residual is perpendicular to b.
Source values may arrive in a different order from the form. Map them explicitly to vector a and target vector b, normalize units, and retain enough precision for the next step after vector projection.
Scaling b by a nonzero number leaves the projection unchanged. Watching this response separates a data-entry mistake from an unexpected but valid value.
If vector a and target vector b are exact counts, more result digits may be meaningful than when they are measured approximations. Let the least certain source guide the final presentation.
Dot product supplies the alignment factor while subtraction gives the residual. This page deliberately reports only the former interpretation.
Label the output as vector projection in notes or tables. Store vector a and target vector b beside it when the result will be reused in a later stage.
The source values for Vector Projection are Vector a, and Target vector b. Subtract the projection from vector a. That residual should have a zero dot product with the target vector b.
For Vector Projection, write the component dimension beside the answer before combining it with later work.
The projection of a onto b is ((a·b)/(b·b))b and lies along b.
It decomposes forces, velocities, errors, shadows, and least-squares components.
Projection onto the zero vector is undefined, and reversing b does not change the projected vector.
Find the scalar component factor, multiply b, and check that the residual is perpendicular to b.