Vector Projection Calculator
Project vector u onto vector v in 2D or 3D: get the projection vector, the scalar projection or the projection's length.
Last updated: October 2026
Projection of u onto v
(0.8, 0, 1.6)
Formula below · 2 sources (mathworld.wolfram.com, Wikipedia) · Updated Oct 2026
Compare with similar
About this calculator
The projection of u onto v is the part of u that points along v, like the shadow u casts on the line through v.
proj_v(u) = (u · v ÷ |v|²) × v
u · v = u_x v_x + u_y v_y + u_z v_z is the dot product and |v|² = v_x² + v_y² + v_z². The number u · v ÷ |v|² scales v to give the projection vector.
The scalar projection, u · v ÷ |v|, is the signed length of that shadow: positive if u points partly the same way as v, negative if it points partly the opposite way. The length of the projection is its absolute value.
For 2D vectors, leave the z components blank or at 0. v must not be the zero vector, since it has no direction to project onto. The part of u at right angles to v is u − proj_v(u); in physics this splits a force into components along and across a surface.
How to use
Enter the components of u and v and choose what to show. The default u = (2, 3, 1) and v = (1, 0, 2) has u · v = 2 + 0 + 2 = 4 and |v|² = 1 + 0 + 4 = 5, so proj_v(u) = 0.8 × (1, 0, 2) = (0.8, 0, 1.6). The scalar projection is 4 ÷ √5 ≈ 1.79. In 2D, u = (3, 4) onto v = (2, 1) gives u · v = 10 and |v|² = 5, so the projection is (4, 2) and the scalar projection is 10 ÷ √5 ≈ 4.47.
Frequently asked questions
What is the difference between scalar and vector projection?
The scalar projection is a single signed number, u · v ÷ |v|, giving how far u reaches along v. The vector projection is that length pointed in v's direction: (u · v ÷ |v|²) v.
Why must v be nonzero?
The formula divides by |v|², which is 0 for the zero vector, and the zero vector has no direction to project onto.
How is projection used in physics?
To split a force into the part along a direction and the part perpendicular to it. Work done by a force is the scalar projection of the force onto the displacement times the distance moved.
Related calculators
Sources & references
Think a result is wrong? Report a problem with this calculator.