Vector Projection Calculator
Calculate the projection of one vector onto another, finding the component of the first vector that lies along the direction of the second.
This free online vector projection calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.
x component of the vector to project
y component of the vector to project
x component of the vector to project onto
y component of the vector to project onto
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Vector Projection Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.
Review your inputs
Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.
Read the results
The Vector Projection Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.
Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
Formula Reference
Vector Projection Calculator Formula
See calculator inputs for the governing equation
Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.
When to Use This Calculator
- •Use the Vector Projection Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
- •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
- •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
- •Use it to explore the behavior of mathematical functions across a range of inputs.
About This Calculator
The Vector Projection Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Calculate the projection of one vector onto another, finding the component of the first vector that lies along the direction of the second. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.
About Vector Projection Calculator
Vector projection determines the component of one vector that lies along the direction of another vector. Given vectors A and B, the projection of A onto B produces a new vector that points in the direction of B and has a length equal to the shadow that A casts onto B. This operation is fundamental in physics for decomposing forces, in computer graphics for lighting calculations, and in linear algebra for constructing orthogonal bases via the Gram-Schmidt process. The scalar projection gives the signed length of this shadow, while the vector projection gives the full vector result. When the scalar projection is positive, A has a component in the direction of B; when negative, the component is in the opposite direction. This calculator computes both the scalar factor and the resulting vector projection components for two-dimensional vectors, providing a complete decomposition of how one vector relates to the direction of another.
The Math Behind It
Formula Reference
Vector Projection
proj_B(A) = ((A·B) / (B·B)) × B
Variables: A = vector being projected; B = vector projected onto
Worked Examples
Example 1: Project a vector onto the x-axis
Project A = (3, 4) onto B = (1, 0).
The projection of (3, 4) onto (1, 0) is (3, 0) with magnitude 3.
Example 2: Projection onto a diagonal vector
Project A = (4, 2) onto B = (1, 1).
The projection is (3, 3) with magnitude approximately 4.24.
Common Mistakes & Tips
- !Dividing by |B| instead of B·B — the formula uses the squared magnitude, not the magnitude itself.
- !Confusing scalar projection (a number) with vector projection (a vector) — the scalar is the factor, the vector includes direction.
- !Projecting onto the zero vector, which is undefined since B·B = 0.
- !Reversing the order: proj_B(A) and proj_A(B) are different operations with different results.
Related Concepts
Used in These Calculators
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Frequently Asked Questions
What is the difference between scalar and vector projection?
The scalar projection is a single number representing the signed length of A's shadow on B. The vector projection is a full vector that has this length and points in the direction of B. Multiply the scalar projection by the unit vector of B to get the vector projection.
Can the projection be longer than the original vector?
No. The magnitude of the projection is |A|cos(θ), which is at most |A| since |cos(θ)| ≤ 1. The projection is always shorter than or equal to the original vector.
What does a negative scalar projection mean?
A negative scalar projection means the angle between A and B is obtuse (greater than 90 degrees), so A has a component in the opposite direction of B.