Vector Calculator
Add, subtract, and find the dot product, cross product, angle, projection, magnitude, unit vector and norms of 2-D or 3-D vectors.
Change the values and press Calculate to work out your own figures.
Vector addition
Enter the vector components (leave z blank for 2-D).
Show the working
- Add the components
Vector subtraction
Enter the vector components (leave z blank for 2-D).
Show the working
- Subtract the components
Dot product
Enter the vector components (leave z blank for 2-D).
Show the working
- a · b = 3 × 1 + 4 × 2 = 11
Cross product
Enter the vector components (leave z blank for 2-D).
Show the working
- a × 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)
Angle between two vectors
Enter the vector components (leave z blank for 2-D).
Show the working
- cos θ = (a · b) ÷ (|a| |b|)
Projection of a onto b
Enter the vector components (leave z blank for 2-D).
Show the working
- proj_b a = ((a · b) ÷ |b|²) b = 11/5 b
Magnitude
Enter the vector components (leave z blank for 2-D).
Show the working
- |a| = √(3² + 4²) = √25
Unit vector
Enter the vector components (leave z blank for 2-D).
Show the working
- â = a ÷ |a|
Vector norms
Enter the vector components (leave z blank for 2-D).
Show the working
- ‖a‖₁ = Σ|aᵢ|; ‖a‖₂ = √Σaᵢ²; ‖a‖∞ = max|aᵢ|
How to use it
Enter the vector components (leave z blank for 2-D). Enter the vector components (leave z blank for 2-D). Enter the vector components (leave z blank for 2-D). Enter the vector components (leave z blank for 2-D). Enter the vector components (leave z blank for 2-D). Enter the vector components (leave z blank for 2-D). Enter the vector components (leave z blank for 2-D). Enter the vector components (leave z blank for 2-D). Enter the vector components (leave z blank for 2-D).
Enter coordinates as whole numbers, decimals or fractions (3/4). Show the working gives the formula with your numbers.
Key facts
a · b = Σ aᵢbᵢ; |a| = √(a · a)
2-D and 3-D
Leave the z-components blank for plane vectors; the cross product treats them as lying in the plane z = 0.
Questions
What is (3, 4) · (1, 2)?
11.
What is |(3, 4)|?
5.
Formulas
a · b = Σ aᵢbᵢ; |a| = √(a · a)
Sources
Limitations
- Vectors have two or three components (leave z blank for 2-D).
- Earth distances assume the WGS 84 ellipsoid (sea level).