Vector Calculator
Perform 2D and 3D vector operations: addition, subtraction, dot product, cross product (3D only), magnitude, unit vector, and angle between two vectors.
Vector A
Vector B
Vector Operation Formulas
Magnitude: |A| = √(x² + y² + z²)
Unit vector: Â = A / |A|
Dot product: A·B = axbx + ayby + azbz (scalar)
Cross product: A×B = (aybz−azby, azbx−axbz, axby−aybx) (3D only)
Angle: θ = arccos(A·B / (|A||B|))
| Relationship | Condition |
|---|---|
| Parallel vectors | Cross product = zero vector |
| Perpendicular vectors | Dot product = 0 |
| Same direction | Dot product > 0 |
| Opposite direction | Dot product < 0 |
How to use
Dot product
A scalar value equal to |A||B|cos(θ). If the dot product is 0, the vectors are perpendicular. If positive, the angle between them is less than 90°.
Cross product (3D only)
Returns a vector perpendicular to both A and B. Its magnitude equals |A||B|sin(θ). Used in physics for torque, angular momentum, and finding surface normals.
Unit vector
A vector with magnitude 1 pointing in the same direction as A. Found by dividing each component by the magnitude.
Angle
Calculated from the dot product formula: θ = arccos(A·B / (|A||B|)). Returns both degrees and radians.
Formula
Magnitude: |A| = √(x² + y² + z²)
Dot: A·B = axbx + ayby + azbz
Cross: A×B = (aybz−azby, azbx−axbz, axby−aybx)
Angle: θ = arccos(A·B / |A||B|)
Tips
- •Two vectors are parallel if their cross product is the zero vector.
- •Two vectors are perpendicular (orthogonal) if their dot product is 0.
- •The cross product is not commutative: A×B = −(B×A).