Vector Calculator
What is Vector Calculator?
A vector is a mathematical object that has both magnitude (length) and direction. Vector calculations include algebraic operations (addition, subtraction, scalar multiplication), geometric interpretations (dot product, cross product, angle, projection), normalization, and basic vector calculus (velocity, acceleration, curvature) in 2D, 3D, and n-dimensional space.
The Vector Calculator is a fast, accurate online tool that instantly performs all standard vector operations in 2D, 3D, and nD space, including addition, dot product, cross product, magnitude, unit vector, angle between vectors, scalar/vector projection, linear dependence check, and introductory vector calculus operations (velocity, acceleration, curvature). It supports Cartesian, polar, and spherical coordinates with automatic conversion. Perfect for vector calculator online, vector operations calculator, dot product calculator, cross product calculator 3D, vector projection tool, magnitude and unit vector calculator, linear dependence checker, vector calculus calculator, position velocity acceleration vectors, and linear algebra homework help.
This vector calculator provides relevant visualizations (vector arrows, parallelogram law, projection lines, angle arcs, 3D isometric projection, dynamic SVG diagrams), a dedicated section for comments, analysis and recommendations, full step-by-step calculation with every component shown, CSV export/download of results (all components, scalars, booleans, properties), and a Colorblind view for accessibility considerations.
How to use Vector Calculator
Purpose: Perform exact algebraic and geometric vector operations, check properties (orthogonality, parallelism, linear independence), and compute basic vector calculus quantities for any number of dimensions.
Inputs you will enter:
- Vector A (array: [x, y] for 2D, [x, y, z] for 3D, or longer for nD)
- Vector B (same dimension as A for binary operations)
- Scalar k (for scalar multiplication)
- Operation type (addition, dot, cross, magnitude, angle, projection, etc.)
- Coordinate system (Cartesian default, polar, spherical)
- Optional: time t for vector functions (velocity/acceleration)
Vector Formula
Vector addition \(\mathbf{A} + \mathbf{B} = [A_x + B_x,\ A_y + B_y,\ A_z + B_z]\)
Dot product \(\mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z\)
Cross product (3D) \(\mathbf{A} \times \mathbf{B} = [A_y B_z – A_z B_y,\ A_z B_x – A_x B_z,\ A_x B_y – A_y B_x]\)
Magnitude \(|\mathbf{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}\)
Angle between vectors \(\theta = \cos^{-1} \left( \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|} \right)\)
Where:
- A, B = vectors (components real numbers)
- θ = angle (radians or degrees)
- All operations are component-wise in nD
How to Calculate Vector Operations (Step-by-Step)
- Enter Vector A and Vector B (or scalar k).
- Select operation (addition, dot, cross, magnitude, angle, projection, unit vector, etc.).
- Choose dimension (auto-detected) and coordinate system if needed.
- Calculator validates dimensions, performs the operation component-wise.
- For dot/cross/angle → computes scalar result and geometric interpretation.
- For projection → shows scalar and vector components.
- Checks properties (orthogonal, parallel, zero vector, linear dependence).
- Generates dynamic SVG diagram (arrows, parallelogram, projection, angle arc) and exports full result table.
Examples
Example 1 – 3D Vector Operations A = [3, 4, 0], B = [1, 2, 5] Magnitude |A| = 5 Dot product A·B = 11 Angle θ ≈ 68.9° Cross product A×B = [20, -15, 2] Projection of A onto B = (11/30) B ≈ [0.367, 0.733, 1.833]
Example 2 – Vector Calculus (Parametric Curve) r(t) = [3t², 4t, 5] at t = 2 Velocity v = [12, 4, 0] Speed |v| = √160 ≈ 12.65 Acceleration a = [12, 0, 0] Curvature κ = |a × v| / |v|³ ≈ 0.075 (radius ≈ 13.33)
Vector Operations Categories / Normal Range
| Operation | 2D Result Type | 3D Result Type | Typical Geometric Meaning | Common Use |
|---|---|---|---|---|
| Addition / Subtraction | Vector | Vector | Parallelogram / triangle law | Resultant force |
| Scalar Multiplication | Vector | Vector | Scaling & direction reversal | Force multiples |
| Magnitude | Scalar | Scalar | Length | Speed, distance |
| Dot Product | Scalar | Scalar | Projection, work, orthogonality | Work, angle check |
| Cross Product | Scalar (perp) | Vector | Area, torque, perpendicular vector | Torque, normal |
| Angle Between Vectors | Angle | Angle | 0° parallel, 90° orthogonal | Direction comparison |
| Projection (scalar/vector) | Scalar / Vector | Scalar / Vector | Component along another vector | Component resolution |
| Unit Vector | Vector | Vector | Direction only (length 1) | Unit tangent |
| Linear Dependence Check | Boolean | Boolean | Dependent if rank < number of vectors | Basis check |
Limitations
- Exact symbolic operations only; higher nD or complex expressions use numerical methods.
- Cross product defined only in 3D (2D returns scalar perpendicular magnitude).
- Vector calculus limited to position-velocity-acceleration and basic differential operators.
- No symbolic integration/differentiation for arbitrary vector functions.
- High dimensions (n>20) become computationally heavy in browser.
Disclaimer
This calculator is provided for educational purposes, learning, and mathematical practice only. All final engineering or research applications must be verified with professional software and reviewed by a qualified mathematician or engineer. The developer and platform are not liable for any errors, misinterpretations, or consequences arising from the use of these results in exams, projects, or real-world applications.