Vector Projection Calculator

Category: Linear Algebra
- May 19, 2025
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Calculate the vector projection of one vector onto another. This calculator helps you find the scalar projection, vector projection, and orthogonal components for 2D or 3D vectors.

Vector Dimensions

Vector Input (Cartesian)

Vector A is projected onto Vector B to find the component of A in the direction of B.

Display Settings

Understanding Vector Projections

Vector projection is a fundamental concept in linear algebra and physics. It allows us to find the component of one vector in the direction of another vector, which is essential for analyzing forces, velocities, and other vector quantities.

Key Concepts
  • Scalar Projection: The length of the shadow cast by one vector onto another. It can be positive or negative depending on the angle between vectors.
  • Vector Projection: A vector in the direction of the target vector with magnitude equal to the scalar projection.
  • Orthogonal Component: The component of the source vector that is perpendicular to the target vector.
  • Dot Product: A mathematical operation that gives the scalar projection multiplied by the magnitude of the target vector.
Mathematical Formulas
  • Scalar Projection: projBA = (A·B)/|B| = |A|·cos(θ)
  • Vector Projection: projBA = ((A·B)/|B|2)·B
  • Orthogonal Component: A⊥ = A - projBA
  • Dot Product (2D): A·B = AxBx + AyBy
  • Dot Product (3D): A·B = AxBx + AyBy + AzBz
Applications
  • Physics: Analyzing component forces, work done by a force, and friction
  • Computer Graphics: Shadow calculations, lighting effects, and collision detection
  • Engineering: Stress analysis, fluid dynamics, and electrical circuit analysis
  • Machine Learning: Feature projection, dimensionality reduction, and principal component analysis
  • Signal Processing: Fourier transforms and filtering operations

Disclaimer: This calculator provides mathematical results for educational and reference purposes. While it's designed to be accurate, always verify critical calculations using multiple methods, especially for scientific, engineering, or professional applications.

Scalar Projection Formula:
projBA = (A · B) / |B| = |A| · cos(θ)

Vector Projection Formula:
projBA = ((A · B) / |B|²) × B

Orthogonal Component:
A⊥ = A - projBA

What Is the Vector Projection Calculator?

The Vector Projection Calculator is an interactive tool that helps you understand how one vector projects onto another in both 2D and 3D space. It calculates three key components:

  • Scalar Projection: The length of vector A in the direction of vector B.
  • Vector Projection: A vector representing how much of A points in the direction of B.
  • Orthogonal Component: The portion of vector A that lies perpendicular to B.

This tool is particularly useful for students, educators, and professionals working in fields like Physics, engineering, and computer graphics. Whether you're studying forces, resolving components, or visualizing vector interactions, this calculator saves time and reduces errors.

How to Use the Calculator

Follow these steps to use the Vector Projection Calculator effectively:

  • Select the vector dimension: 2D (x, y) or 3D (x, y, z).
  • Choose your input method: Cartesian coordinates or Magnitude & Angle (2D only).
  • Enter the values for Vector A (source) and Vector B (target).
  • Adjust display options like decimal places, detailed calculations, and diagrams.
  • Click the "Calculate Projection" button to see the results instantly.

Why This Calculator Is Helpful

This calculator offers clear insights into vector behavior by showing how vectors relate geometrically and numerically. Some common benefits include:

  • Breaking down forces in mechanical systems.
  • Understanding how much of a force or motion is effective in a certain direction.
  • Learning and teaching vector math with visual and step-by-step explanations.
  • Validating manual calculations in academic or engineering tasks.

Real-World Applications

The Vector Projection Calculator has broad applications in various domains:

  • Physics: Work done, force components, motion analysis.
  • Engineering: Load distribution, structural analysis, system modeling.
  • Computer Graphics: Lighting, shading, and object alignment.
  • Machine Learning: Feature projection and dimensionality reduction.

Frequently Asked Questions (FAQ)

Can I use this for 3D vectors?
Yes. Choose the 3D option to input x, y, and z components for both vectors.

What’s the difference between scalar and vector projection?
Scalar projection gives a single number (how far along B vector A extends), while vector projection returns a new vector in the direction of B.

What happens if Vector B is a zero vector?
The calculator will not process the input. Vector B must have a magnitude greater than zero because it defines the direction of projection.

Why is this useful in learning Linear Algebra?
It helps visualize and understand dot products, projections, and orthogonal components, which are foundational concepts in vector math and linear transformations.

Explore Related Tools

This calculator complements several Other linear algebra tools like the Vector Addition Calculator for combining vectors, the Dot Product Calculator for finding scalar products, and the Vector Magnitude Calculator for measuring vector lengths.

It also fits well alongside matrix-focused tools such as the Matrix Inverse Calculator, QR Factorization Calculator, and LU Decomposition Calculator, which help with more advanced transformations and linear system solutions.