Vector Length & Operations Calculator
Calculate the length (magnitude) of 2D and 3D vectors, dot product, and cross product. Full step-by-step formulas included.
Vector a
Vector b
How to Use the Vector Length & Operations Calculator
- Enter the numbers or values in the input fields.
- The result is calculated and displayed automatically.
- Review the step-by-step solution or detailed breakdown.
- Copy the result or adjust inputs for a new calculation.
Referência Rápida
| De | Para |
|---|---|
| 2 + 3 | 5 |
| 12 × 12 | 144 |
| √144 | 12 |
| 2¹⁰ | 1,024 |
| π | 3.14159 |
| e | 2.71828 |
Casos de Uso
- •Checking homework or exam answers quickly and accurately.
- •Verifying manual calculations in professional or academic work.
- •Learning mathematical concepts with instant visual feedback.
- •Performing quick computations during meetings or presentations.
Fórmula
Vector length: |v| = √(x² + y² + z²). Dot product: a·b = a₁b₁ + a₂b₂ + a₃b₃. Cross product (3D only): a×b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁).
Perguntas Frequentes
What is vector length?
The length (or magnitude) of a vector is its Euclidean norm — the distance from the origin to the point it represents. For a 2D vector (x, y) it is √(x² + y²).
What is the dot product?
The dot (scalar) product of two vectors is the sum of the products of their corresponding components. It equals |a|·|b|·cos(θ), where θ is the angle between the vectors.
What is the cross product?
The cross product of two 3D vectors produces a new vector perpendicular to both inputs. Its magnitude equals |a|·|b|·sin(θ), the area of the parallelogram spanned by the vectors.