# Vector Projection Calculator

> Calculate vector projections with interactive 2D/3D visualizations

**Category:** Utility
**Keywords:** calculator, tool
**URL:** https://complete.tools/vector-projection-calculator

## How it calculates

The formula for the projection of vector A onto vector B is given by: 

proj_B(A) = (A • B / |B|^2) × B

where: 
- A is the vector being projected, represented as A = (A_x, A_y, A_z) 
- B is the vector onto which A is being projected, represented as B = (B_x, B_y, B_z) 
- A • B is the dot product of vectors A and B, calculated as A_x × B_x + A_y × B_y + A_z × B_z 
- |B| is the magnitude of vector B, calculated as √(B_x² + B_y² + B_z²) 
- proj_B(A) is the resulting projection vector. 

This relationship shows that the projection is scaled by the ratio of the dot product of A and B to the square of the magnitude of B, effectively capturing the component of A in the direction of B.

## Who should use this

1. Physicists analyzing forces acting on an object in different directions. 
2. Computer graphics developers calculating light vector projections for shading models. 
3. Civil engineers determining the vector components of forces acting on structural elements. 
4. Data scientists performing dimensionality reduction techniques in high-dimensional data.

## Worked examples

Example 1: Project vector A = (3, 4) onto vector B = (1, 2). 
First, calculate the dot product: A • B = 3 × 1 + 4 × 2 = 3 + 8 = 11. 
Next, calculate the magnitude of B: |B| = √(1² + 2²) = √(1 + 4) = √5. 
Then, square the magnitude: |B|² = 5. 
Now, find the projection: proj_B(A) = (11 / 5) × (1, 2) = (2.2, 4.4). 

Example 2: Project vector A = (1, 0, 2) onto vector B = (0, 1, 0). 
Calculate the dot product: A • B = 1 × 0 + 0 × 1 + 2 × 0 = 0. 
Magnitude of B: |B| = √(0² + 1² + 0²) = 1. 
Square of the magnitude: |B|² = 1. 
Projection: proj_B(A) = (0 / 1) × (0, 1, 0) = (0, 0, 0). This shows that A has no component in the direction of B.

## Limitations

1. The tool assumes the vectors are provided in Cartesian coordinates; other coordinate systems may require conversion. 
2. The precision of the calculations is limited by the floating-point representation in the software, which may introduce rounding errors for very large or very small vectors. 
3. The tool may not handle edge cases where vectors are zero-length, as division by zero occurs in the magnitude calculation. 
4. Visualizations may become cluttered or less interpretable when projecting numerous vectors simultaneously, affecting clarity.

## FAQs

**Q:** How does the tool handle vector normalization during projection calculations?


**A:** The tool does not explicitly normalize vectors during the projection calculation; it uses the original vector lengths in the formula. Normalization can be performed beforehand if necessary.


**Q:** Can the tool project vectors in different dimensions?


**A:** The tool is specifically designed for 2D and 3D vectors. Attempting to project vectors in dimensions other than these will yield inaccurate results.


**Q:** What happens if I input non-numeric values for vectors?


**A:** The tool requires numeric input for vector components. Non-numeric values will result in an error, as calculations cannot proceed without valid numerical data.


**Q:** Is the tool capable of visualizing projections in non-Euclidean spaces?


**A:** The tool is limited to Euclidean space projections. Non-Euclidean projections require different mathematical frameworks not supported by this tool.

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*Generated from [complete.tools/vector-projection-calculator](https://complete.tools/vector-projection-calculator)*