Vector Projection Calculator

Visualize and calculate the projection of one vector onto another with this interactive tool.

Input Vectors

Vector A (e.g., [1, 2, 3]):

Vector B (e.g., [4, 5, 6]):

Result

Projection Vector:

Calculation Steps

Vector Visualization

Understanding Vector Projection

Vector projection is a fundamental concept in linear algebra that helps us understand how much of one vector lies in the direction of another. Specifically, the projection of vector A onto vector B (also known as the vector component of A along B) gives us a vector that is parallel to B and represents the "shadow" of A on B.

Formula

The formula for projecting vector A onto vector B is given by:

$$ \text{proj}_\mathbf{B} \mathbf{A} = \frac{\mathbf{A} \cdot \mathbf{B}}{\mathbf{B} \cdot \mathbf{B}} \mathbf{B} $$

A and B are vectors.
A · B represents the dot product of A and B.
B · B is the dot product of B with itself, which is the square of the magnitude of B, i.e., |B|².

Use Cases

In physics, to find the component of a force in a certain direction.
In computer graphics, for lighting calculations and shadow rendering.
In machine learning, for feature extraction and dimensionality reduction techniques like Principal Component Analysis (PCA).

Learn more about vector projection on Wikipedia.