Vector Projection Calculator

Welcome to the Vector Projection Calculator, a powerful linear algebra tool that computes the projection of one vector onto another with step-by-step formula breakdowns, interactive geometric visualization, and orthogonal decomposition. Whether you are studying linear algebra, working on physics problems, or analyzing data in machine learning, this calculator makes vector projections intuitive and easy to understand.

What is Vector Projection?

Vector projection is a fundamental operation in linear algebra that finds how much of one vector goes in the direction of another. Given vectors a and b, the projection of a onto b produces a new vector that lies along b and represents the "shadow" of a cast onto the line defined by b.

There are two related concepts:

• Scalar projection (component): A single number representing the signed length of the projection along b
• Vector projection: A vector that lies along b with the magnitude equal to the scalar projection

Vector Projection Formula

Scalar Projection Formula

Orthogonal Decomposition

Any vector a can be decomposed into two perpendicular components relative to b:

Where \(\vec{a}_{\perp} = \vec{a} - \text{proj}_{\vec{b}} \vec{a}\) is the component of a perpendicular to b (also called vector rejection).

How to Use This Calculator

1. Choose dimension: Select 2D or 3D vectors using the toggle buttons.
2. Enter vectors: Input the components of vector a (the vector being projected) and vector b (the direction of projection).
3. Calculate: Click "Calculate Projection" to see the full results including vector projection, scalar projection, orthogonal component, angle between vectors, and step-by-step solution.
4. Explore visualization: Review the interactive diagram showing all vectors and the geometric relationship between them.

Understanding Your Results

• Vector Projection: The projection vector that lies along b
• Scalar Projection: The signed length of the projection (positive if angle < 90°, negative if angle > 90°)
• Orthogonal Component: The part of a perpendicular to b
• Angle Between Vectors: The angle θ in both degrees and radians
• Projection Scalar (a·b/b·b): The multiplier applied to b to get the projection vector

Applications of Vector Projection

Special Cases

• Parallel vectors (θ = 0°): The projection of a onto b equals a itself (scaled by magnitude ratio)
• Anti-parallel vectors (θ = 180°): The projection points in the opposite direction of b
• Perpendicular vectors (θ = 90°): The projection is the zero vector — a has no component along b
• Scalar projection = 0: The vectors are orthogonal
• Negative scalar projection: The angle between vectors exceeds 90°

Frequently Asked Questions

What is a vector projection?

A vector projection of a onto b is the component of a that lies in the direction of b. It is calculated as proj_b(a) = (a·b / b·b) × b. The result is a vector that points in the same (or opposite) direction as b, representing how much of a goes along b.

What is the difference between scalar projection and vector projection?

Scalar projection gives a single number representing the signed length of the projection along b, calculated as comp_b(a) = a·b / |b|. Vector projection gives a vector result that has both magnitude and direction, calculated as proj_b(a) = (a·b / b·b) × b. The scalar projection is the magnitude (with sign) of the vector projection.

What is the orthogonal component (vector rejection)?

The orthogonal component (also called vector rejection) is the part of vector a that is perpendicular to vector b. It is calculated as a_⊥ = a − proj_b(a). Together, the projection and rejection decompose a into two perpendicular components whose sum equals the original vector.

Can the scalar projection be negative?

Yes. A negative scalar projection means the angle between the two vectors is greater than 90°, so vector a has a component pointing in the opposite direction of b. The absolute value of the scalar projection still represents the length of the projected shadow.

Why is vector projection important in machine learning?

Vector projection is fundamental to techniques like Principal Component Analysis (PCA), which projects high-dimensional data onto the directions of maximum variance. It is also used in regression (projecting response vectors onto feature spaces), recommendation systems, and dimensionality reduction, making it one of the most widely used operations in data science.

Additional Resources

• Vector Projection - Wikipedia
• Dot Product - Wikipedia