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.
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.
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.
Keep the Vector Projection row or coordinate order for Vector a. Read Target vector b in the same Vector Projection order.
Check one Vector Projection component by hand. Substitute or multiply Target vector b back to verify Vector Projection.
Check Vector Projection from Vector a, then verify Target vector b. Estimate the Vector Projection Vector projection before computing it again. If Target vector b changes during Vector Projection, keep Target vector b fixed. That Vector Projection comparison shows whether Vector projection moves as expected.
Substitute the calculated Vector projection into the defining Vector Projection relationship when that reversal is possible. The recovered value should agree with Vector a under Target vector b. A mismatch points to rounding, entry order, or an assumption involving Target vector b.
Units and conventions belong with a Vector Projection answer. Confirm that Vector a and Target vector b use the intended interpretation, then label Vector projection the same way. A numerically correct Vector projection can still answer the wrong Vector Projection question when that context changes.
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.