Vector Projection Calculator
Vector projection decomposes one vector into the component parallel to another vector, a common step in geometry, physics and engineering calculations.
How to use this calculator
- Enter the x, y and z components for both vectors. The result reports both the projected vector and scalar projection.
- Check the units and definitions before comparing results with another source.
- Change one input at a time when comparing scenarios, and keep the reporting period and assumptions consistent.
Formula used by the vector projection calculator
projᵦ(a) = ((a · b)/(b · b)) b; scalar projection = (a · b)/|b|
- projᵦ(a) = ((a · b)/(b · b)) b; scalar projection = (a · b)/|b|
- The target vector must not be the zero vector. All components must represent compatible dimensions and units.
- Inputs are checked for required values and valid ranges; displayed results are rounded for readability.
Worked examples
Worked example
Project A = (3, 4, 0) onto B = (1, 0, 0).
The dot product is 3 and |B|² = 1, so the projection vector is (3, 0, 0).
The dot product is 3 and |B|² = 1, so the projection vector is (3, 0, 0).
Understanding the result
Vector projection decomposes one vector into the component parallel to another vector, a common step in geometry, physics and engineering calculations.
The target vector must not be the zero vector. All components must represent compatible dimensions and units.
Important notes
The target vector must not be the zero vector. All components must represent compatible dimensions and units.
The result applies to the numbers and assumptions entered here; verify specifications for safety-critical or regulated uses.
Frequently asked questions
What does a projection represent?
It is the part of the first vector that lies along the direction of the second vector.
Can the projected scalar be negative?
Yes. A negative scalar projection means the first vector points partly opposite the chosen target direction.
How do I use this calculator?
Enter the x, y and z components for both vectors. The result reports both the projected vector and scalar projection.
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Related explanations
Learn the method behind the number or compare the terminology before using the calculator.
- Area — Understand area as the amount of two-dimensional surface inside a shape and review common formulas.
This page provides a calculation tool plus explanatory information so you can understand what the result means and when the underlying method may not fit your situation.