Vector & Dot Product Calculator
Calculate vector operations including dot product, cross product, magnitude, and angles
Vector Operations
Vector A
Vector B
How to Use the Vector Calculator
Vector Notation
- 2D Vectors: Represented as (x, y) with two components
- 3D Vectors: Represented as (x, y, z) with three components
- Components: Enter decimal numbers (positive or negative)
- Zero Vector: All components equal to zero
Vector Operations
- Dot Product: A · B = |A||B|cos(θ), gives scalar result
- Cross Product: A × B, perpendicular to both vectors
- Magnitude: |A| = √(x² + y² + z²), length of vector
- Angle: θ = arccos((A · B)/(|A||B|)), between 0° and 180°
- Addition: A + B = (A₁+B₁, A₂+B₂, A₃+B₃)
- Unit Vector: Â = A/|A|, direction with magnitude 1
Key Properties
- Dot Product: Positive for acute angles, negative for obtuse, zero for perpendicular
- Cross Product: Magnitude equals parallelogram area, direction by right-hand rule
- Unit Vector: Always has magnitude 1, preserves direction
- Commutative: A · B = B · A, but A × B = -(B × A)
Applications
- Physics: Force, velocity, displacement calculations
- Engineering: Structural analysis, moment calculations
- Computer Graphics: 3D transformations, lighting calculations
- Navigation: Direction and distance calculations
Frequently Asked Questions
What's the difference between dot and cross products?
Dot product gives a scalar (number) and measures how much vectors point in the same direction. Cross product gives a vector perpendicular to both input vectors and measures the area of the parallelogram they form.
Why is cross product different in 2D vs 3D?
In 2D, cross product gives only the z-component of what would be the 3D result. In 3D, you get a full vector with three components. The 2D result is often called a "pseudo-scalar."
What does a zero dot product mean?
A zero dot product means the vectors are perpendicular (orthogonal). The angle between them is exactly 90°. This is useful for testing if vectors are at right angles.
How do I find the angle between vectors?
Use the formula θ = arccos((A · B)/(|A||B|)). The dot product divided by the product of magnitudes gives the cosine of the angle, then take the inverse cosine.
What is a unit vector used for?
Unit vectors have magnitude 1 and show pure direction. They're used to normalize vectors, define coordinate systems, and perform direction-based calculations without considering magnitude.
Related Calculators
- Matrix Calculator - For linear transformations and vector spaces
- Trigonometric Calculator - For angle calculations in vector problems
- Distance Calculator - For vector displacement calculations
- Force Calculator - For physics applications of vectors