Skalarprodukt-Rechner

Skalarprodukt-Rechner

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

Dot Product vs. Cross Product

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

1. Select the dimension: Choose 2D, 3D, 4D, or Custom for higher dimensions. Click a quick example to auto-fill sample values.
2. Enter Vector a: Type the components separated by commas (e.g., 3, 4, 5 for a 3D vector).
3. Enter Vector b: Type the components of the second vector in the same dimension.
4. Watch the live preview: The vector diagram updates in real-time as you type, showing the spatial relationship and angle between vectors.
5. 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

Lineare Algebra:

• Determinantenrechner
• Eigenwert- und Eigenvektor-Rechner
• matrixrechner
• Partialbruchzerlegungsrechner
• vektorrechner
• Gram-Schmidt-Rechner
• Vektorprojektion Rechner
• Matrix-LU-Zerlegung-Rechner
• Singulärwertzerlegung (SVD) Rechner
• Matrixrang-Rechner
• Matrixspur-Rechner
• Jacobi-Matrix-Rechner Neu
• RREF Rechner (Zeilenstufenform) Neu
• Matrix Inverse Rechner Neu
• Matrix-Multiplikationsrechner Neu
• Skalarprodukt-Rechner Neu
• Kreuzprodukt-Rechner Neu
• Vektorbetrag-Rechner Neu
• Einheitsvektor-Rechner Neu
• Winkel zwischen Vektoren Rechner Neu
• Nullraum-Rechner Neu
• Spaltenraum-Rechner Neu
• Cramersche Regel Rechner Neu
• Matrix-Diagonalisierung-Rechner Neu
• QR-Zerlegung Rechner Neu
• Cholesky-Zerlegung-Rechner Neu
• Matrixpotenz-Rechner Neu
• Charakteristisches Polynom Rechner Neu
• Jordansche Normalform Rechner Neu
• Matrixexponential-Rechner Neu
• Tensorprodukt-Rechner Neu