Vector Projection Calculator

Introduction

The projection of one vector onto another describes how vector $A$ resolves along the direction of vector $B$ . This relationship is defined through scalar and vector components derived from the dot product, allowing the parallel and orthogonal contributions of $A$ relative to $B$ to be quantified. This section outlines the fundamental parameters required for analysing directional decomposition within linear algebra and coordinate geometry.

What this calculator does

Performs the decomposition of two 2D vectors entered by the user. It requires the Cartesian coordinates for vector $A$ and vector $B$ . The system then computes the dot product, scalar projection, and vector projection. Additionally, it calculates the vector rejection to identify the orthogonal component and determines the precise angle between the two vectors in degrees and radians.

Formula used

The vector projection ${proj}_{B} A$ is derived using the dot product $A \cdot B$ and the magnitude squared of $B$ . The scalar projection represents the signed length of the projection. The vector rejection $R$ is the difference between the original vector and its projection.

{proj}_{B} A = (\frac{A \cdot B}{{| B |}^{2}}) B

{comp}_{B} A = \frac{A \cdot B}{| B |}

How to use this calculator

1. Enter the x and y coordinates for Vector A.
2. Enter the x and y coordinates for Vector B.
3. Select the preferred number of decimal places for the output precision.
4. Execute the calculation to view the numerical results and the interactive plot.

Example calculation

Scenario: Analysing the directional components of a force vector relative to a displacement axis in a theoretical physics simulation involving particle motion along a plane.

Inputs: Vector $A$ is $(3, 4)$ and Vector $B$ is $(5, 0)$ .

Working:

Step 1: $A \cdot B = (3 \times 5) + (4 \times 0)$

Step 2: $A \cdot B = 15$

Step 3: ${| B |}^{2} = 5^{2} + 0^{2} = 25$

Step 4: $P = (15 / 25) \times (5, 0)$

Result: Vector Projection P = (3, 0).

Interpretation: The vector A has a component of 3 units acting in the same direction as vector B.

Summary: The calculation successfully decomposes vector A into its parallel projection on B.

Understanding the result

The scalar projection indicates the magnitude of vector A in the direction of vector B; a negative result suggests the vectors point in opposite directions. The vector rejection identifies the part of A that is perfectly perpendicular to B, revealing the orthogonal distance between the tip of A and the line defined by B.

Assumptions and limitations

The calculation assumes all vectors exist within a 2D Euclidean space. It requires that vector $B$ has a non-zero magnitude, as projection onto a point is mathematically undefined. Coordinate values are limited to a specific range to maintain numerical stability.

Common mistakes to avoid

A frequent error is confusing the vector projection with the scalar projection, or assuming that the order of vectors does not matter. Projecting B onto A yields different results than projecting A onto B. Additionally, ensure vector B is not a zero vector $(0, 0)$ to avoid division errors.

Sensitivity and robustness

The calculation is stable for most inputs but becomes highly sensitive when the magnitude of vector $B$ is extremely small, approaching $10^{- 18}$ . In such cases, small changes in coordinates can lead to large fluctuations in the projection factor due to the inverse square relationship with magnitude.

Troubleshooting

If an error appears, verify that both coordinates for vector B are not zero. If results look unusual, check if the decimal precision is set too low for very small vectors. Ensure no HTML tags or non-numeric characters have been included in the input fields, as these will trigger validation errors.

Frequently asked questions

What is vector rejection?

Vector rejection is the component of the first vector that is orthogonal to the second vector, calculated by subtracting the projection from the original vector.

Can the scalar projection be negative?

Yes, the scalar projection is negative if the angle between the two vectors is obtuse, specifically between 90 and 180 degrees.

What happens if the vectors are perpendicular?

If the vectors are perpendicular, their dot product is zero, resulting in a vector projection of (0, 0).

Where this calculation is used

This mathematical process is essential in various academic disciplines. In linear algebra, it is used for the Gram-Schmidt process to create orthogonal bases. In computer graphics, vector projection helps determine shadows and lighting reflections on surfaces. Environmental researchers use it to analyse wind velocity components relative to a specific path, while sports analysts might use it to decompose directional forces in athlete movement. It is a cornerstone of mathematical modelling where directional data must be standardised or compared against a reference axis.

Results are based on standard mathematical and statistical methods and may involve rounding or approximation. If precise accuracy is required, please verify results independently. See full disclaimer.