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The projection of one vector onto another answers a simple geometric question: how much of vector A points in the same direction as vector B? It's a foundational operation used throughout physics, engineering, and computer graphics.
This calculator finds both the scalar projection (a single number) and the vector projection (a full 2D vector) of vector A onto vector B, with a complete step-by-step solution.
How it works
Enter the x and y components of vectors A and B. The calculator computes the dot product of A and B, divides by the magnitude of B to find the scalar projection, and scales vector B by the appropriate factor to find the full vector projection.
- Enter vector a — x (vector to project).
- Enter vector a — y.
- Enter vector b — x (direction to project onto).
- Enter vector b — y.
- Click Calculate to see your results.
Examples
Projecting (3, 4) onto (5, 0)
Projecting vector A = (3, 4) onto vector B = (5, 0) (which points purely along the x-axis) gives a scalar projection of 3 and a vector projection of (3, 0) — exactly A's x-component, since B points straight along the x-axis.
Who should use it
- Finding the component of a force acting along a specific direction in physics problems.
- Computing shadow/projection effects in computer graphics and game development.
Industry applications
- Physics and engineering education
- Computer graphics and game development
Advantages
- Calculates both the scalar and full vector projection in one step.
- Includes a complete, formula-based step-by-step solution.
Limitations
- Limited to 2D vectors — 3D vector projection requires an extended formula.
Common mistakes to avoid
- Confusing the scalar projection (a number) with the vector projection (a vector) — they answer related but different questions.
- Forgetting that the order matters: projecting A onto B is generally different from projecting B onto A.
Best practices
- Use the scalar projection when you just need a single "how much overlap" number, and the vector projection when you need the actual resulting vector for further calculations.
Tips
- If you only need the "how much overlap" number and not the resulting vector, the scalar projection alone is usually simpler to work with.