1. What is Vector Projection?

Vector projection is a fundamental operation in linear algebra that finds the component of one vector (A) that lies along the direction of another vector (B). The projection represents the "shadow" or component of vector A in the direction of vector B.

2. How Does the Calculator Work?

The calculator uses the projection formula:

Where:

• \( A \cdot B \) — Dot product of vectors A and B
• \( B \cdot B \) — Dot product of vector B with itself (squared magnitude)
• \( B \) — The vector being projected onto

Explanation: The formula calculates how much of vector A points in the direction of vector B, scaled appropriately to match B's magnitude.

3. Applications of Vector Projection

Details: Vector projection has applications in physics (decomposing forces), computer graphics (shading, reflections), machine learning (dimensionality reduction), and signal processing (filtering unwanted components).

4. Using the Calculator

Tips: Enter vectors as comma-separated values (e.g., "1,2,3" for a 3D vector). Both vectors must have the same dimension. The calculator will compute the projection of vector A onto vector B.

5. Frequently Asked Questions (FAQ)

Q1: What if vectors have different dimensions?
A: The calculator requires both vectors to have the same number of components. If dimensions differ, the projection cannot be computed.

Q2: What is the geometric interpretation of projection?
A: The projection represents the closest point on the line spanned by B to the tip of vector A, when their tails are placed at the origin.

Q3: What happens if vector B is the zero vector?
A: Division by zero occurs since B·B = 0. The projection is undefined for the zero vector.

Q4: Can I project onto higher-dimensional subspaces?
A: This calculator only projects onto a single vector (1D subspace). Projection onto planes or higher-dimensional subspaces requires different techniques.

Q5: How is this related to orthogonal decomposition?
A: The projection gives the parallel component. The perpendicular component can be found by subtracting the projection from the original vector: A - proj_B(A).