Angle Between Vectors Calculator
Calculate the angle between two 2D or 3D vectors using the dot product formula cos(θ) = (a·b)/(|a||b|). Get step-by-step solutions, results in both degrees and radians, interactive vector diagram, and geometric interpretation.
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About Angle Between Vectors Calculator
The Angle Between Vectors Calculator finds the angle between two 2D or 3D vectors using the dot product formula \(\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\). Enter your vector components to instantly get the angle in both degrees and radians, a complete step-by-step solution, vector magnitudes, dot product, unit vectors, projection, geometric interpretation, and an interactive diagram with toggleable layers.
The Dot Product Angle Formula
The angle \(\theta\) between two vectors \(\vec{a}\) and \(\vec{b}\) is derived from the dot product identity:
$$\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| \cdot |\vec{b}|}$$
Where:
- \(\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + \ldots + a_n b_n\) is the dot product
- \(|\vec{a}| = \sqrt{a_1^2 + a_2^2 + \ldots + a_n^2}\) is the magnitude of vector a
- \(\theta = \arccos\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\right)\) gives the angle between 0° and 180°
Understanding the Dot Product Sign
Real-World Applications
Key Formulas
| Formula | Expression | Description |
|---|---|---|
| Dot Product (2D) | \(a_1 b_1 + a_2 b_2\) | Sum of component-wise products |
| Dot Product (3D) | \(a_1 b_1 + a_2 b_2 + a_3 b_3\) | Extends to three components |
| Magnitude | \(|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}\) | Length (norm) of a vector |
| Angle | \(\theta = \arccos\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\right)\) | Always between 0° and 180° |
| Cosine Similarity | \(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\) | Same as cos θ — ranges from −1 to 1 |
| Projection | \(\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\vec{b}\) | Component of a along b |
How to Use the Angle Between Vectors Calculator
- Enter Vector a: Type the components separated by commas. Use 2 components for 2D (e.g., 3, 4) or 3 components for 3D (e.g., 1, 2, 3). Click any quick example to auto-fill both fields.
- Enter Vector b: Type the components of the second vector in the same dimension as vector a.
- Watch the live preview: The diagram updates in real time, showing both vectors and the computed angle as you type.
- Click Calculate: Press the button to get the full result including angle in degrees and radians, step-by-step solution, all related quantities, and the interactive diagram.
- Explore the diagram: Toggle layers (angle arc, projection, grid, labels) for different visualizations. For 3D vectors, drag to rotate the view.
2D vs 3D Vectors
The dot product angle formula works identically in both 2D and 3D — only the number of components changes. In 2D, vectors have components (x, y) and the diagram shows a flat Cartesian plane with a clear angle arc. In 3D, vectors have components (x, y, z) and the diagram provides an interactive rotatable isometric view. The mathematical principle is the same: compute the dot product, divide by the product of magnitudes, and take the arccosine.
FAQ
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"Angle Between Vectors Calculator" at https://MiniWebtool.com// from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: 2026-04-10
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