Orthogonal Projection Calculator

Category: Linear Algebra
- May 15, 2025
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This calculator helps you compute the orthogonal projection of one vector onto another vector. It calculates the projection vector, scalar projection, and related components.

Vector Inputs

x:
y:
z:
x:
y:
z:

Understanding Orthogonal Projection

Orthogonal projection is a fundamental concept in linear algebra and geometry. It involves projecting one vector onto another, resulting in a new vector that represents the component of the first vector in the direction of the second.

Key Concepts
  • Vector Projection: The resulting vector when projecting vector A onto vector B
  • Scalar Projection: The magnitude of the projection vector (a scalar value)
  • Parallel Component: The component of vector A that is parallel to vector B
  • Perpendicular Component: The component of vector A that is perpendicular to vector B
Formula

The projection of vector A onto vector B is given by:

projBA = (A·B / |B|²) × B

Where:

  • A·B is the dot product of vectors A and B
  • |B| is the magnitude (length) of vector B
  • |B|² is the square of the magnitude of vector B
Applications
  • Physics: Resolving forces into components parallel and perpendicular to a reference direction
  • Computer Graphics: Shadow calculations and finding the closest point on a line
  • Machine Learning: Feature projection and dimensionality reduction
  • Engineering: Analyzing structural forces and moments

Disclaimer: This calculator provides mathematical results based on vector algebra principles. It's designed for educational purposes and should not replace proper mathematical analysis for critical applications.

Orthogonal Projection Calculator: Purpose and Benefits

The Orthogonal Projection Calculator is a visual and interactive tool for calculating the projection of one vector onto another in both 2D and 3D space. This operation is essential in vector mathematics, especially when analyzing components of force, direction, or movement. The calculator is especially helpful in educational, engineering, and Physics contexts where vector decomposition is required.

By entering the components of two vectors—Vector A (the one being projected) and Vector B (the target)—the tool computes and displays:

  • Projection Vector: The vector representation of A projected onto B
  • Scalar Projection: The length of the projection
  • Parallel and Perpendicular Components: Understanding how A splits relative to B
  • Angle Between Vectors: Calculated in degrees for intuitive insight

Orthogonal Projection Formula:

projBA = (A · B / |B|²) × B

Where:

  • A · B is the dot product of vectors A and B
  • |B| is the magnitude of vector B
  • projBA is the projection vector of A onto B

How to Use the Calculator

  1. Select the vector dimension: choose between 2D and 3D space.
  2. Input the x, y (and optionally z) components of Vector A and Vector B.
  3. Click the Calculate Projection button to view the results.
  4. Review the visual chart and step-by-step breakdown provided below the input section.
  5. To start over, press the Reset button.

Why Orthogonal Projection Matters

Orthogonal projections are used in a variety of fields to break down vectors into understandable parts:

  • Physics: Separating force vectors into useful directions
  • Computer Graphics: Rendering shadows and projections
  • Engineering: Resolving loads in structural analysis
  • Machine Learning: Reducing vector dimensions for efficiency

FAQ: Common Questions

Q: Can I use this for 3D vectors?
Yes. Simply select "3D (x, y, z)" from the dimension menu to enable z-component inputs.

Q: What if Vector B is zero?
The tool will alert you. Projection onto a zero vector is undefined and cannot be calculated.

Q: How does this compare with Other vector tools?
This calculator complements other tools like the Vector Addition Calculator, Dot Product Calculator, and Unit Vector Calculator by focusing specifically on the projection operation. It shares foundational principles with matrix-based tools such as the QR Factorization Calculator and Matrix Inverse Calculator, which are also based on vector manipulation and decomposition.

Related Tools for Deeper Analysis

Once you're comfortable with vector projections, you might find these tools useful for advancing your understanding of vector and matrix operations:

Whether you're solving linear systems, analyzing forces, or preparing for exams, this Orthogonal Projection Calculator is a practical and accessible step in understanding vector behavior and interactions.