Vector Projection Calculator

Your details

Switch between 2-component and 3-component vectors.
x-component of vector A.
y-component of vector A.
z-component of vector A (3D only).
x-component of vector B (the direction to project onto).
y-component of vector B.
z-component of vector B (3D only).
Scalar projection of A onto BObtuse angle
-4.0972

Signed length of the projection - negative when the angle is obtuse.

Projection vector - x-1.5738
Projection vector - y-3.1475
Projection vector - z2.0984
Orthogonal component - x3.5738
Orthogonal component - y0.1475
Orthogonal component - z2.9016
Dot product (A . B)-32
|A| (magnitude of A)6.1644
|B| (magnitude of B)7.8102
Angle between A and B131.66deg
Projection magnitude4.0972
Vectors orthogonal?No
Scalar projection-4.0972
Projection magnitude4.0972
|A|6.1644
|B|7.8102

Scalar projection: -4.0972, vector projection: (-1.5738, -3.1475, 2.0984)

  • The angle is 131.66 degrees (obtuse), so the scalar projection is negative - A points partly away from B.
  • The projection captures 66.5% of the length of A along the direction of B.
  • The projection vector and orthogonal component together reconstruct A exactly: proj + orth = A (in 3D).

Next stepIn physics, the scalar projection gives the component of a force, velocity, or displacement in a chosen direction. In data science it is the basis of PCA and linear regression projections.

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