Dot Product Calculator
Calculate the dot product (scalar product) of two vectors in 2D, 3D, or higher dimensions. Get the angle between vectors, magnitudes, scalar and vector projections, geometric interpretation, and step-by-step formulas with an interactive vector diagram.
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About Dot Product Calculator
The Dot Product Calculator computes the scalar product of two vectors in 2D, 3D, or higher dimensions using the algebraic formula \(\vec{a} \cdot \vec{b} = \sum_{i=1}^{n} a_i b_i\). Enter the components of your two vectors to instantly get the dot product, angle between vectors, magnitudes, scalar and vector projections, geometric interpretation, and a step-by-step solution with an interactive vector diagram.
Real-World Applications
Key Formulas
| Property | Formula | Description |
|---|---|---|
| Dot Product | \(\vec{a} \cdot \vec{b} = \sum a_i b_i\) | Sum of component-wise products |
| Geometric Form | \(\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta\) | Product of magnitudes times cosine of angle |
| Angle | \(\theta = \arccos\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\) | Angle between the two vectors (0° to 180°) |
| Magnitude | \(|\vec{a}| = \sqrt{\sum a_i^2}\) | Length (Euclidean norm) of a vector |
| Scalar Projection | \(\text{comp}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}\) | Signed length of a's shadow on b |
| Vector Projection | \(\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\vec{b}\) | Vector component of a along b |
Dot Product vs. Cross Product
Dot Product (a · b)
Produces a scalar value. Works in any dimension (2D, 3D, nD). Measures how much two vectors point in the same direction. Zero when vectors are perpendicular. Used for projections, angles, and work calculations.
Cross Product (a × b)
Produces a vector perpendicular to both inputs. Only defined in 3D (and 7D). Magnitude equals the area of the parallelogram formed by the vectors. Zero when vectors are parallel. Used for torque, normals, and area calculations.
Understanding the Geometric Interpretation
The dot product has a deep geometric meaning: \(\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta\). This tells us:
- Positive dot product (θ < 90°): vectors point in a generally similar direction.
- Zero dot product (θ = 90°): vectors are perpendicular (orthogonal) — this is the foundation of orthogonality tests in linear algebra.
- Negative dot product (θ > 90°): vectors point in generally opposite directions.
The scalar projection of \(\vec{a}\) onto \(\vec{b}\) gives the signed length of \(\vec{a}\)'s "shadow" when light shines perpendicular to \(\vec{b}\). The vector projection gives this shadow as an actual vector along \(\vec{b}\).
How to Use the Dot Product Calculator
- Select the dimension: Choose 2D, 3D, 4D, or Custom for higher dimensions. Click a quick example to auto-fill sample values.
- Enter Vector a: Type the components separated by commas (e.g., 3, 4, 5 for a 3D vector).
- Enter Vector b: Type the components of the second vector in the same dimension.
- Watch the live preview: The vector diagram updates in real-time as you type, showing the spatial relationship and angle between vectors.
- Click Calculate: Press the button to get the full results including dot product, angle, magnitudes, projections, interpretation, and step-by-step formulas.
Properties of the Dot Product
- Commutative: \(\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}\)
- Distributive: \(\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}\)
- Scalar multiplication: \((k\vec{a}) \cdot \vec{b} = k(\vec{a} \cdot \vec{b})\)
- Self dot product: \(\vec{a} \cdot \vec{a} = |\vec{a}|^2\) (square of the magnitude)
- Cauchy-Schwarz inequality: \(|\vec{a} \cdot \vec{b}| \leq |\vec{a}||\vec{b}|\)
FAQ
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"Dot Product Calculator" at https://MiniWebtool.com/dot-product-calculator/ from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: 2026-04-09
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