Quick answer
The projection of a onto b is proj_b a = ((a · b) ÷ (b · b)) b, and the scalar projection is (a · b) ÷ |b|. Projecting (3, 4) onto (1, 0) gives (3, 0), with a scalar projection of 3.
Enter vectors A and B. The calculator returns the projection vector and the scalar projection.
Related tools: Vector Magnitude Calculator: Length (Euclidean norm) of a vector. Unit Vector Calculator: Unit vector in the direction of a vector. Vector Subtraction Calculator: Subtract one vector from another.
How to use the Vector Projection Calculator
- Type vector A, for example 1, 2, 3.
- Type vector B, for example 1, 2, 3.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Vector A | Text, such as 1, 2, 3 | — |
| Vector B | Text, such as 1, 2, 3 | — |
Formula and method
Formula
proj_b a = ((a · b) ÷ (b · b)) b
IEEE-754 double precision; shown to at most 6 decimal places.
b must be non-zero.
Worked examples
This example is verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Vector A = 3, 4; Vector B = 1, 0.
| Result | Value |
|---|---|
| Projection | (3, 0) |
| Scalar projection | 3 |
The first line is the headline result (projection). Below it the vector Projection Calculator lists scalar projection, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Geometric meaning
The projection is the 'shadow' of a along the direction of b. Here b points along the x-axis, so the shadow of (3, 4) is its x-part, (3, 0). The perpendicular part, (0, 4), is a − proj.
Uses
Work done by a force along a displacement, resolving forces along a ramp and least-squares fitting. The vector b must be non-zero.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Vector B: entering "0, 0" shows "Vector b must not be zero." beside that field and no result.
Choosing between the Vector Projection Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Vector Magnitude Calculator](/math/vector-magnitude/) | components | Magnitude |
| [Unit Vector Calculator](/math/unit-vector-calculator/) | vector | Unit vector |
| [Vector Subtraction Calculator](/math/vector-subtraction-calculator/) | vector A and vector B | a − b |
| [Vector Calculator](/math/vector/) | vector a and vector b | Sum (a + b) |
| [Slope Calculator](/math/slope-calculator/) | x₁, y₁ and x₂ | Slope |
Background: algebra
Many algebra calculators rest on the same ideas: write the problem in a standard form, identify the coefficients, apply the formula and check the answer by substituting it back. For a quadratic ax² + bx + c = 0 the roots are (−b ± √(b² − 4ac)) ÷ 2a, and the discriminant b² − 4ac decides the kind of roots: positive gives two real roots, zero gives one repeated root and negative gives two complex conjugate roots.
If a = 0 the equation is linear, not quadratic. A sign slip on the −b term shifts both roots by the same amount, so substitute your roots back to check.
Assumptions and limitations
- Non-zero b
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for vector B to confirm this.
Frequently asked questions
What is the projection of a on itself?
Projecting a vector onto itself returns the vector, since its component along its own direction is the whole vector.
What if a and b are perpendicular?
The projection is zero.
What is the scalar projection?
The signed length of the shadow.
What does the Vector Projection Calculator calculate?
Projection of vector a onto vector b. It returns Projection and Scalar projection.
What formula does the Vector Projection Calculator use?
proj_b a = ((a · b) ÷ (b · b)) b. b must be non-zero.
What do I need to enter in the vector Projection Calculator?
You enter Vector A and Vector B.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Non-zero b.
What is an example of a vector Projection result?
With vector A 3, 4 and vector B 1, 0, the projection is (3, 0).
Related calculators
- Vector Magnitude Calculator — Length (Euclidean norm) of a vector.
- Unit Vector Calculator — Unit vector in the direction of a vector.
- Vector Subtraction Calculator — Subtract one vector from another.
- Vector Calculator — Sum, difference, dot product, cross product and angle of two vectors.
- Slope Calculator — Slope, angle, distance and line equation through two points.
- LCM Calculator — Least common multiple of two or more numbers.
Sources
- OpenStax, Precalculus 2e (Rice University): vectors, polar coordinates and trigonometry
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.