What is a Vector Calculator?
A vector calculator is a mathematical tool that performs operations on vectors - quantities that have both magnitude (size) and direction. Unlike regular numbers (scalars), vectors are represented as ordered sets of components, such as (x, y) for 2D or (x, y, z) for 3D space.
Supported Operations
Operations Requiring Two Vectors
- Addition (A + B) - Combines two vectors by adding their corresponding components
- Subtraction (A − B) - Finds the difference between two vectors
- Dot Product (A · B) - Calculates the scalar product, useful for finding angles and projections
- Cross Product (A × B) - Produces a vector perpendicular to both input vectors (3D only)
- Angle Between Vectors - Determines the angle between two vectors in degrees and radians
- Projection - Projects one vector onto another
Operations on Single Vector
- Magnitude (|A|) - Calculates the length of a vector
- Unit Vector (Â) - Normalizes a vector to length 1 while preserving direction
- Scalar Multiplication (k · A) - Scales a vector by a constant factor
2D and 3D Support
2D Vectors
Work with two-dimensional vectors using x and y components for planar calculations.
- Perfect for 2D geometry
- Simpler visualization
- All operations except cross product
3D Vectors
Extend to three-dimensional space with x, y, and z components for spatial calculations.
- Full 3D space operations
- Cross product available
- Physics and engineering applications
Simply toggle between dimensions using the 2D/3D switch. Note that the cross product operation is only available in 3D mode, as it requires three-dimensional space to produce a meaningful result.
- 1. What is a Vector Calculator?
- 2. How to Use the Vector Calculator
- 3. Features
- 4. Frequently Asked Questions
- 4.1. What is the difference between dot product and cross product?
- 4.2. Why is cross product only available in 3D mode?
- 4.3. What is a unit vector?
- 4.4. Why can't I normalize a zero vector?
- 4.5. What does vector projection mean?
- 4.6. How is the angle between vectors calculated?
- 4.7. What precision does the calculator use?
- 4.8. Can I use negative numbers?
How to Use the Vector Calculator
Choose Your Dimension
Start by selecting whether you want to work with 2D or 3D vectors using the dimension toggle at the top. In 2D mode, you'll enter x and y components. In 3D mode, an additional z component field appears.
Enter Vector Values
Input the components of your vectors in the provided fields:
- Vector A - Enter x, y (and z for 3D) values in the blue-labeled card
- Vector B - Enter values in the red-labeled card (visible for two-vector operations)
Select an Operation
Choose from the available operations organized in two groups:
- Two Vectors - Operations requiring both A and B (add, subtract, dot product, cross product, angle, projection)
- Single Vector - Operations on vector A only (magnitude, unit vector, scalar multiplication)
View Results
Results appear instantly as you enter values or change operations:
- Vector results display as coordinates (x, y) or (x, y, z)
- Scalar results (dot product, magnitude) display as a single number
- Angle results show both degrees and radians
Using the Visualization
The coordinate plane on the right shows your vectors graphically:
Vector A
Vector B
Result Vector
Use the zoom controls (+, −, reset) to adjust the view scale and see vectors of different magnitudes clearly.
Viewing Formulas and Steps
Click "Show formula & steps" below the result to see:
- The mathematical formula used for the calculation
- Step-by-step breakdown of how the result was computed
Features
Real-Time Calculations
Results update instantly as you type. There's no need to click a calculate button - simply enter your values and see the result immediately.
- Instant feedback
- No calculate button needed
- Experiment freely with values
Interactive Visualization
Built-in coordinate plane provides visual representation of your vectors with color-coded arrows and adjustable zoom.
- Vectors drawn as arrows from origin
- Color-coded for easy identification
- Coordinates displayed at arrow tips
- Zoom controls for different magnitudes
Step-by-Step Solutions
Every calculation includes a detailed breakdown showing the formula, intermediate steps, and how the final result was derived.
- Mathematical formulas
- Intermediate calculation steps
- Clear derivation process
Comprehensive Operations
Covers all essential vector operations from basic arithmetic to advanced geometric calculations.
- Addition and subtraction
- Dot and cross products
- Angles and projections
- Magnitude and normalization
Scalar Multiplication Presets
Quick-select buttons provide common scalar values for instant vector scaling operations.
- −1 - Reverse direction
- 2 - Double length
- 0.5 - Halve length
- 10 - Scale up 10×
Dark Mode Support
Automatically adapts to your system's dark mode preference with optimized colors for both themes.
- Auto theme detection
- Comfortable viewing
- Optimized visualization colors
Copy Results
Click the copy button to copy the result to your clipboard. Vector results are copied in coordinate format (x, y) or (x, y, z), making it easy to paste into other applications or documents for further use.
Frequently Asked Questions
What is the difference between dot product and cross product?
The dot product (A · B) returns a scalar (single number) and measures how much two vectors point in the same direction. It equals |A||B|cos(θ), where θ is the angle between them.
The cross product (A × B) returns a vector perpendicular to both input vectors, with magnitude |A||B|sin(θ). Cross product is only defined for 3D vectors.
Scalar Result
- Returns a single number
- Measures alignment
- Works in 2D and 3D
- Formula: A · B = |A||B|cos(θ)
Vector Result
- Returns a vector
- Perpendicular to both inputs
- 3D only
- Magnitude: |A||B|sin(θ)
Why is cross product only available in 3D mode?
The cross product produces a vector perpendicular to both input vectors. In 2D space, there's no third dimension for this perpendicular vector to exist in.
What is a unit vector?
A unit vector has a magnitude (length) of exactly 1 while maintaining the same direction as the original vector. It's calculated by dividing each component by the vector's magnitude:
 = A / |A|
Example:
A = (3, 4)
|A| = √(3² + 4²) = 5
 = (3/5, 4/5) = (0.6, 0.8)Unit vectors are useful for representing directions without regard to magnitude, commonly used in physics and computer graphics.
Why can't I normalize a zero vector?
A zero vector (0, 0) or (0, 0, 0) has no direction - it's just a point. Since normalization preserves direction while setting magnitude to 1, it's mathematically undefined for a vector with no direction.
What does vector projection mean?
The projection of A onto B (projB(A)) gives the component of vector A that lies along the direction of vector B.
Imagine shining a light perpendicular to B - the shadow of A on B is the projection.
proj_B(A) = ((A · B) / |B|²) × B
Where:
• A · B is the dot product
• |B|² is the magnitude of B squared
• Result is a vector along B's directionHow is the angle between vectors calculated?
The angle θ between vectors A and B is found using the dot product formula:
θ = arccos((A · B) / (|A| × |B|))
Result formats:
• Degrees: 0° to 180°
• Radians: 0 to πThe calculator displays the result in both degrees and radians for convenience.
What precision does the calculator use?
Results are displayed with up to 4 decimal places, with trailing zeros removed for cleaner output. Internal calculations use full floating-point precision to ensure accuracy.
Can I use negative numbers?
Yes, all vector components can be positive, negative, or zero. Negative values simply indicate direction along the negative axis.
Example: The vector (−3, 4) points left and up in 2D space, while (3, −4) points right and down.
- Negative x: Points left (west)
- Negative y: Points down (south)
- Negative z: Points backward (in 3D)
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