Vector Projection Calculator
Calculate the projection of vector a onto vector b, including scalar projection, vector projection, and rejection.
Vectors
Vector a (to be projected)
|a| = 5.0000
Vector b (projection direction)
|b| = 5.0000
a · b = 15.0000
Angle = 53.13°
Vector Projection (proj_b a)
(3.0000, 0.0000, 0.0000)
|proj| = 3.0000
Scalar Projection (comp_b a)
3.000000
The signed length of the projection
Orthogonal Rejection
(0.0000, 4.0000, 0.0000)
|rej| = 4.0000
Component of a perpendicular to b
Decomposition
a = proj_b(a) + rej_b(a)
(3.0000, 4.0000, 0.0000) = (3.0000, 0.0000, 0.0000) + (0.0000, 4.0000, 0.0000)
Projection Ratio
60.00%
of |a| in direction of b
Unit Vector b̂
(1.0000, 0.0000, 0.0000)
Projection Formulas
Scalar Projection
comp_b(a) = (a · b) / |b|
= |a| cos(θ)
Vector Projection
proj_b(a) = ((a · b) / |b|²) × b
= (a · b̂) × b̂