How to Use Vector Calculator
The Vector Calculator performs all common vector operations in one place, with step-by-step calculation details shown for every result.
- Select Dimension — Choose 2D for two-dimensional vectors (x, y) or 3D for three-dimensional vectors (x, y, z) in the Vector Calculator.
- Choose Operation — Pick the vector operation from the dropdown: addition, subtraction, dot product, cross product, magnitude, unit vector, angle, or projection.
- Enter Vector A — Input the x, y, (and z) components of the first vector.
- Enter Vector B or Scalar — For binary operations provide Vector B components; for scalar multiplication enter the scalar value. The Vector Calculator hides irrelevant inputs automatically.
- Read Results — The Vector Calculator displays the result, its magnitude (for vector outputs), and the step-by-step breakdown.
Formula & Theory — Vector Calculator
The Vector Calculator implements standard linear-algebra vector formulas:
Addition: A + B = (ax+bx, ay+by, az+bz)
Subtraction: A - B = (ax-bx, ay-by, az-bz)
Scalar mult: kA = (k·ax, k·ay, k·az)
Dot product: A · B = ax·bx + ay·by + az·bz
Cross product: A × B = (ay·bz - az·by, az·bx - ax·bz, ax·by - ay·bx)
Magnitude: |A| = √(ax² + ay² + az²)
Unit vector: Â = A / |A|
Angle: θ = arccos((A · B) / (|A|·|B|))
Projection: proj_B(A) = (A·B / |B|²) · B
| Symbol | Meaning |
|---|---|
| A, B | Input vectors |
| k | Scalar multiplier |
| θ | Angle between A and B (degrees) |
| Â | Unit vector in direction of A |
The Vector Calculator uses exact floating-point arithmetic. Angles are clamped to the range [−1, 1] before arccos to avoid numerical errors near 0° and 180°.
Cross Product Note
The cross product is only geometrically meaningful in 3D. In 2D mode, the Vector Calculator returns the z-component of the 3D cross product, which equals the signed area of the parallelogram formed by A and B.
Use Cases for Vector Calculator
The Vector Calculator is useful in a variety of situations across physics, engineering, and mathematics:
- Physics — Forces, velocities, and accelerations are vectors. Use the Vector Calculator to find net force (addition), work done (dot product), or torque (cross product).
- Computer Graphics & Game Development — Normal vectors, lighting models, and collision detection all require the cross product and dot product operations provided by the Vector Calculator.
- Navigation & GPS — Bearing calculations, displacement vectors, and velocity decomposition are standard applications of the Vector Calculator in geospatial contexts.
- Machine Learning — Cosine similarity (derived from the dot product and magnitudes) is computed by the Vector Calculator's angle function, making it useful for NLP and recommendation systems.
- Structural Engineering — Moment of a force about a point is the cross product of the position vector and force vector — a direct Vector Calculator application.
Whether you are solving a physics problem, building a 3D application, or studying linear algebra, the Vector Calculator provides accurate results and clear step-by-step explanations.