Vector calculator

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How to Use This Vector Calculator

Select vector dimension - Choose between 2D (x, y) or 3D (x, y, z) vectors
Enter vector components - Input the values for each vector component
Choose operation - Select from addition, subtraction, dot product, cross product, or other operations
Get instant results - View the answer with detailed step-by-step calculations

Vector Operations Available

Addition

Add two or more vectors component-wise

\vec{a} + \vec{b} = (a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3})

Subtraction

Subtract one vector from another

\vec{a} - \vec{b} = (a_{1} - b_{1}, a_{2} - b_{2}, a_{3} - b_{3})

Dot Product

Calculate scalar product of vectors

\vec{a} \cdot \vec{b} = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}

Cross Product

Find perpendicular vector (3D only)

\vec{a} \times \vec{b} = (a_{2} b_{3} - a_{3} b_{2}, a_{3} b_{1} - a_{1} b_{3}, a_{1} b_{2} - a_{2} b_{1})

Magnitude

Calculate vector length

| \vec{a} | = \sqrt{a_{1}^{2} + a_{2}^{2} + a_{3}^{2}}

Normalize

Convert vector to unit vector (length 1)

\vec{û} = \frac{\vec{u}}{| \vec{u} |}

Angle Between

Find angle between two vectors

θ = \cos^{- 1} (\frac{\vec{a} \cdot \vec{b}}{| \vec{a} | | \vec{b} |})

Projection

Project one vector onto another

{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{{| \vec{b} |}^{2}} \vec{b}

Detailed Examples

Example 1: 2D Vector Addition

Given vectors a = (3, 4) and b = (1, 2)

Solution:

a + b = (3+1, 4+2) = (4, 6)

Example 2: 3D Dot Product

Given vectors a = (2, 3, 1) and b = (1, 4, 2)

Solution:

a · b = (2×1) + (3×4) + (1×2) = 2 + 12 + 2 = 16

Example 3: Cross Product

Given vectors a = (1, 2, 3) and b = (4, 5, 6)

Solution:

a × b = ((2×6)-(3×5), (3×4)-(1×6), (1×5)-(2×4))
= (12-15, 12-6, 5-8) = (-3, 6, -3)

Example 4: Vector Magnitude and Unit Vector

Given vector v = (3, 4, 0)

Magnitude:

|v| = √(3² + 4² + 0²) = √(9 + 16) = √25 = 5

Unit Vector:

v̂ = v/|v| = (3/5, 4/5, 0) = (0.6, 0.8, 0)

Properties of Vector Operations

Property	Addition/Subtraction	Dot Product	Cross Product
Commutative	✓ a + b = b + a	✓ a · b = b · a	✗ a × b = -(b × a)
Associative	✓ (a + b) + c = a + (b + c)	✗ Not applicable	✗ Not associative
Distributive	✓ k(a + b) = ka + kb	✓ a · (b + c) = a · b + a · c	✓ a × (b + c) = a × b + a × c
Identity Element	✓ Zero vector (0, 0, 0)	✗ No identity	✗ No identity

Special Types of Vectors

Zero Vector

A vector with all components equal to zero: 0 = (0, 0, 0)

Magnitude is zero
No specific direction
Identity element for vector addition

Unit Vector

A vector with magnitude equal to 1

Used to represent direction only
Standard unit vectors: î = (1,0,0), ĵ = (0,1,0), k̂ = (0,0,1)

Position Vector

A vector from the origin to a point in space

Represents the location of a point
Components are the coordinates of the point

Orthogonal Vectors

Two vectors are orthogonal (perpendicular) if their dot product is zero

a ⊥ b if a · b = 0

Parallel Vectors

Two vectors are parallel if one is a scalar multiple of the other

a ∥ b if a = kb for some scalar k

Real-World Applications

Physics and Engineering

Force and Motion: Calculating resultant forces, velocity, and acceleration
Electromagnetic Fields: Electric and magnetic field calculations
Mechanics: Torque calculations using cross product
Work and Energy: Work = Force · Displacement

Computer Graphics and Gaming

3D Rendering: Camera positioning and object transformations
Lighting: Surface normal calculations for shading
Physics Engines: Collision detection and response
Animation: Character movement and rotations

Navigation and GPS

Direction Finding: Calculating bearing and heading
Distance Calculations: Great circle distances on Earth
Route Planning: Optimal path calculations

Machine Learning and Data Science

Feature Vectors: Representing data points in high-dimensional space
Similarity Measures: Cosine similarity using dot product
Neural Networks: Weight vectors and gradient calculations
Principal Component Analysis: Finding orthogonal basis vectors

Vector Representation in Different Coordinate Systems

Cartesian Coordinates

Standard (x, y, z) representation

v = xî + yĵ + zk̂

Polar Coordinates (2D)

Magnitude and angle representation

x = r cos(θ), y = r sin(θ)

Spherical Coordinates (3D)

Radius, polar angle, and azimuthal angle

x = r sin(φ) cos(θ)
y = r sin(φ) sin(θ)
z = r cos(φ)

💡 Pro Tips for Using This Calculator

Check dimensions: Ensure vectors have the same dimension for addition/subtraction
Cross product: Only defined for 3D vectors
Angle calculation: Result is in degrees by default
Precision: Adjust decimal places in settings for more accurate results
Multiple vectors: Chain operations for calculations with 2 vectors
Export formats: Download results as PNG, HTML or MathML

Common Mistakes to Avoid

Mixing dimensions: Cannot add a 2D vector to a 3D vector
Order in cross product: Remember a × b ≠ b × a
Division by zero: Check for zero magnitude when normalizing
Angle units: Ensure you're using the correct unit (degrees vs radians)
Component order: Maintain consistent (x, y, z) ordering

Frequently Asked Questions

What's the difference between dot product and cross product?

Dot product returns a scalar (single number) and measures how much vectors point in the same direction. Cross product returns a vector perpendicular to both input vectors (3D only).

Can I calculate cross product for 2D vectors?

Cross product is only defined for 3D vectors. For 2D vectors, you can treat them as 3D with z=0, resulting in a vector pointing along the z-axis.

How do I know if two vectors are perpendicular?

Two vectors are perpendicular (orthogonal) if their dot product equals zero.

What does vector magnitude represent?

Magnitude represents the length or size of the vector. In physics, it could be the strength of a force or the speed of velocity.

How do I find the angle between two vectors?

Use the formula: θ = cos⁻¹(a·b / |a||b|). This calculator computes it automatically.

What's a unit vector used for?

Unit vectors represent pure direction without magnitude. They're useful for specifying directions in physics and computer graphics.

Quick Formula Reference

                Essential Vector Formulas
                📐 Magnitude: |v| = √(x² + y² + z²)
➕ Addition: a + b = (a₁+b₁, a₂+b₂, a₃+b₃)
➖ Subtraction: a - b = (a₁-b₁, a₂-b₂, a₃-b₃)
• Dot Product: a · b = a₁b₁ + a₂b₂ + a₃b₃
✕ Cross Product: a × b = (a₂b₃-a₃b₂, a₃b₁-a₁b₃, a₁b₂-a₂b₁)
📊 Angle: θ = cos⁻¹(a·b / |a||b|)
↗️ Unit Vector: û = u/|u|
📈 Projection: proj_b(a) = ((a·b)/|b|²)b

            