Unit & Physics

Free 2D & 3D Vector Math Calculator — Dot Product, Cross Product & Magnitude

Vectors are essential mathematical tools in physics, computer graphics, structural mechanics, and navigation. Performing multi-dimensional vector operations like cross products, dot products, and unit vector normalizations by hand requires meticulous calculations. Our free vector calculator provides fast, precise computations for 2D and 3D vector space operations. Use our cross product calculator step by step and vector magnitude solver 3d to compute vector sums, scalar products, vector products, magnitudes, and vector angles in seconds.

2D & 3D Vector Math Calculator

Calculate vector addition, dot product, cross product, magnitude, unit vectors & angles

Vector A Components

A_x

A_y

A_z

Vector B Components

B_x

B_y

B_z

Vector Results & Matrix Products

Magnitude |A|:5.0000

Magnitude |B|:5.4772

Vector Addition (A + B):(4, 6, 5)

Vector Subtraction (A - B):(2, 2, -5)

Dot Product (A · B):11

Cross Product (A × B):(20, -15, 2)

Angle θ between A & B:66.32° (1.1575 rad)

Unit Vector Â = (0.6, 0.8, 0)

Mathematical Formulas & Calculation Methods

3D Vector Magnitude Formula |A|

The length of vector A = (ax, ay, az) derived from the 3D Pythagorean theorem.

|A| = √(ax² + ay² + az²)

Dot Product (Scalar Product) A · B

Multiplies corresponding components to produce a scalar number.

A · B = (ax × bx) + (ay × by) + (az × bz)

Cross Product (Vector Product) A × B

Generates a new 3D vector perpendicular to both input vectors.

A × B = (ay·bz - az·by) i - (ax·bz - az·bx) j + (ax·by - ay·bx) k

Angle Between Vectors θ

Calculates the interior angle θ between vectors A and B.

cos(θ) = (A · B) / (|A| × |B|)

Worked Step-by-Step Examples

Example 1: 3D Vector Operations for A = (3, 4, 0) and B = (1, 2, 5)

"Find magnitude |A|, dot product A · B, cross product A × B, and angle θ."

Steps:

Magnitude |A| = √(3² + 4² + 0²) = √(9 + 16) = √25 = 5.0.
Magnitude |B| = √(1² + 2² + 5²) = √(1 + 4 + 25) = √30 ≈ 5.477.
Dot Product A · B = (3 × 1) + (4 × 2) + (0 × 5) = 3 + 8 + 0 = 11.
Cross Product A × B = ((4×5 - 0×2), (0×1 - 3×5), (3×2 - 4×1)) = (20, -15, 2).
Angle cos(θ) = 11 / (5.0 × 5.477) = 11 / 27.386 = 0.4017 → θ = arccos(0.4017) = 66.32°.

Result: |A| = 5, A · B = 11, A × B = (20, -15, 2), θ = 66.32°

What is 2D & 3D Vector Math Calculator?

A unit vector calculator online scales any non-zero vector down to a magnitude of exactly 1.0 while preserving its directional direction.

The dot product indicates whether two vectors are orthogonal (dot product = 0), pointing in similar directions (positive dot product), or pointing in opposite directions (negative dot product).

The cross product applies exclusively in 3D space to calculate normal vectors needed in 3D rendering, lighting shaders, and torque calculations.

How to Use 2D & 3D Vector Math Calculator

1Select dimension: 2D (x, y) or 3D (x, y, z).
2Enter components for Vector A and Vector B.
3Instantly view calculated magnitudes, dot products, cross products, unit vectors, and angle θ.

Key Features

Supports both 2D and 3D Cartesian vector spaces.
Calculates magnitude |A| and |B| step-by-step.
Computes vector addition (A + B) and subtraction (A - B).
Computes scalar dot product (A · B) and vector cross product (A × B).
Calculates unit vectors Â and Angle θ in both degrees and radians.

Benefits of Using 2D & 3D Vector Math Calculator

Saves significant time in physics homework and mechanics engineering problems.

Essential tool for game developers computing 3D lighting normals and vector reflections.

Practical Use Cases

Physics & Statics Students

Calculate force vectors, resultants, torque, and equilibrium moments.

3D Game Developers

Calculate surface normals, directional vectors, and camera ray directions.

Frequently Asked Questions

What is the difference between dot product and cross product?

The dot product results in a scalar (a single number), whereas the cross product (in 3D) results in a new vector that is perpendicular to both original vectors.

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