How to use the vector projection calculator
The projection of vector a onto vector b is the part of a that survives when you drop its shadow along the direction of b. Pick a dimension, type the components, and the calculator returns both the scalar projection and the vector projection. The scalar projection is (a·b)÷|b|, a single signed length along b, while the vector projection multiplies (a·b)÷|b|² by b and comes out as components.
The orthogonal part is a minus the vector projection, so a splits neatly into a projection plus a perpendicular remainder. That split is what you use to resolve a force along and across a ramp, or to get the residual vector in least squares. A negative scalar projection means a points against b.
If b is the zero vector the denominator |b|² becomes zero, so the calculator explains that instead of dividing by zero. Every figure is derived from the unrounded raw values and only rounded to six decimal places for display, and the orthogonal or parallel verdict is decided with an explicit tolerance rather than an exact comparison. Choosing 3D reveals the z component fields.
Frequently asked questions
The scalar projection is one signed length measured along b. The vector projection places that length back along the direction of b, so its magnitude equals the absolute value of the scalar projection.
The dot product is zero, so the two vectors are orthogonal. The projection is also zero when a itself is the zero vector.
The projection formula divides by |b|², so a zero vector b divides by zero and has no direction. At least one component of b must be non-zero.