Vector Calculator
Perform vector operations: dot product, cross product, magnitude, and more.
2D Vectors
3D Vectors
Result
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Enter vectors and select operation
Dot Product
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Cross Product
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|A|
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|B|
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Angle
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A + B
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A - B
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A ⊥ B?
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Vector Operations
Dot Product
A·B = x₁x₂ + y₁y₂ + z₁z₂
The dot product gives a scalar. It measures projection and is zero for perpendicular vectors.
Cross Product (3D only)
A×B = (y₁z₂ - z₁y₂, z₁x₂ - x₁z₂, x₁y₂ - y₁x₂)
The cross product gives a vector perpendicular to both A and B.
Magnitude
|A| = √(x² + y² + z²)
Angle Between Vectors
θ = arccos(A·B / (|A||B|))
Formulas Summary
| Operation | Result Type | Properties |
|---|---|---|
| Dot Product | Scalar | A·B = |A||B|cos(θ) |
| Cross Product | Vector | |A×B| = |A||B|sin(θ) |
| Magnitude | Scalar | Always non-negative |
Frequently Asked Questions
When is the dot product zero?
The dot product is zero when vectors are perpendicular (orthogonal) or one/both vectors are zero.
What does cross product direction mean?
Cross product follows the right-hand rule: curl fingers from A to B, thumb points in A×B direction.