Vector Calculator — Add, Subtract, Dot & Cross Products
Compute vector operations including addition, subtraction, dot product, cross product, magnitudes, and the angle between vectors in 2D or 3D.
Vector Calc
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The Formula
Vector operations are fundamental in physics, engineering, and computer graphics. The vector sum is computed component-wise, while the dot product yields a scalar representing the projection of one vector onto another. The cross product produces a third vector perpendicular to both inputs. The angle between vectors is derived from the dot product formula: cos θ = (a·b)/(|a||b|).
Variable Definitions
First Vector
The first input vector defined by its three spatial components.
Second Vector
The second input vector defined by its three spatial components.
Magnitude (Length)
The length of a vector, calculated as √(x² + y² + z²). Always non-negative.
Dot Product
The scalar product: a₁b₁ + a₂b₂ + a₃b₃. Measures alignment between vectors.
Cross Product
The vector product: a 3D vector perpendicular to both a and b. Magnitude equals area of parallelogram.
How to Use This Calculator
- 1
Enter the x, y, and z components of both vectors. For 2D vectors, leave the z components as 0.
- 2
The calculator computes addition, subtraction, magnitudes, dot product, cross product, and the angle between vectors.
- 3
Review the results: addition and subtraction show component-wise results; the angle shows how aligned the vectors are.
- 4
Vectors with z = 0 are treated as 2D vectors, but cross product results will still be displayed in 3D format.
Quick Reference
| From | To |
|---|---|
| Addition | v₁ + v₂ = (x₁+x₂, y₁+y₂, z₁+z₂) |
| Dot product | a · b = |a||b|cos θ |
| Cross product | |a × b| = |a||b|sin θ |
| Angle | θ = arccos((a·b)/(|a||b|)) |
Common Applications
- Physics: calculating net force by adding force vectors, computing work as the dot product of force and displacement.
- Computer graphics: representing positions, directions, velocities, and normals; performing lighting calculations using dot products.
- Engineering: analyzing structural forces, moments (torque uses cross product), and fluid velocity fields.
- Machine learning: representing feature vectors and computing similarity using dot products in high-dimensional spaces.
- Robotics: describing robot arm positions and orientations, computing transformations between coordinate frames.
Vectors can be added, subtracted, and multiplied using dot and cross products
Understanding the Concept
Vectors are mathematical objects with both magnitude and direction, represented as ordered tuples of numbers. In 2D, a vector has two components (x, y); in 3D, three components (x, y, z). Vector addition is performed component-wise and geometrically follows the parallelogram law. The dot product (or scalar product) produces a scalar from two vectors and is zero when the vectors are perpendicular — making it invaluable for testing orthogonality. The cross product exists only in 3D and produces a vector perpendicular to both inputs, with magnitude equal to the area of the parallelogram they span. The right-hand rule determines the direction of the cross product. The angle between vectors is derived from the dot product formula, and the angle is 0° for parallel vectors, 90° for perpendicular vectors, and 180° for anti-parallel vectors. Vector operations are the mathematical foundation for describing physical quantities like force, velocity, acceleration, and electromagnetic fields.
Frequently Asked Questions
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