2D Vector Projection Calculator

The projection of vector A onto vector B answers: how much of A points in the direction of B?

Scalar projection (component of A along B):

comp = (A dot B) / |B|

This is a signed number. Positive means A and B point in roughly the same direction. Negative means they point in roughly opposite directions. The absolute value is the length of the vector projection.

Vector projection:

proj = (A dot B / |B|^2) x B

This is a vector pointing in the direction of B with magnitude equal to the scalar projection. It is the “shadow” of A cast onto the line through B.

Perpendicular component:

A_perp = A - proj

This is the part of A orthogonal to B. Together, the projection and the perpendicular component reconstruct A: proj + A_perp = A.

Angle between vectors:

theta = arccos( (A dot B) / (|A| x |B|) )

Common uses. In physics, projections decompose forces along and perpendicular to a surface. Finding the component of gravity acting down a ramp is the standard first example. In machine learning, dot products and projections turn up in neural network activations and in principal component analysis (PCA). In computer graphics, a projection is how you work out how much light lands on a surface.

If the scalar projection is zero, the vectors are orthogonal and their dot product is zero. If it equals |A|, then A lies entirely along B.

A worked example

Take A = (3, 4) and B = (1, 0). B points straight along the x-axis, so the projection should simply pick off A’s x-component.

• A · B = 3(1) + 4(0) = 3
• |B| = 1, so the scalar projection is 3/1 = 3
• proj = (3/1²)(1, 0) = (3, 0)
• A_perp = (3, 4) − (3, 0) = (0, 4)
• Check: (3, 0) + (0, 4) = (3, 4) = A ✓

The angle works out at arccos(3 / (5 × 1)) = 53.13°, and cos(53.13°) × 5 = 3, which is the scalar projection again. That is not a coincidence: the scalar projection is |A| cos θ, and the whole subject is that one fact written four different ways.

One thing worth watching. Projecting A onto B is not the same as projecting B onto A. The dot product is symmetric but the divisor is not, so swapping the two inputs gives a different vector unless |A| and |B| happen to be equal.