Education
Vector Calculator
Pick 2D or 3D, pick an operation, and the calculator returns the result. Supports the standard list of vector operations used in geometry, physics, and graphics.
Dimension
Operation
Vector A
Vector B
Dot product A · B
32
Dot product checks orthogonality (zero means perpendicular). Cross product gives a vector perpendicular to both inputs whose magnitude equals |A||B|sin θ.
Examples
(1,2,3) · (4,5,6)
= 32
(1,0,0) × (0,1,0)
= (0, 0, 1)
|(3,4)|
= 5
angle((1,0), (0,1))
= 90°
How it works
Each operation is the standard one taught in linear algebra class. Component-wise for add/subtract/scalar; sums of products for dot; the right-hand-rule formula for cross; Pythagorean for magnitude; division by magnitude for unit vector.
Magnitude · |A| = √(x² + y² + z²)
Dot product · A · B = Σᵢ Aᵢ · Bᵢ
Cross product (3D) · A × B = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)
Angle · cos θ = (A · B) / (|A| · |B|)
Related calculators
- Matrix calculator for matrix operations that act on vectors as columns.
- Angle converter for switching the resulting angle between degrees and radians.
- Right triangle calculator when the question reduces to a right triangle.
- Distance formula calculator for the magnitude of the difference between two points.
- Scientific calculator for arithmetic checks.
- All education calculators.
Frequently asked questions
A vector is an ordered list of numbers that represents a quantity with both magnitude and direction. In 2D it has x and y components; in 3D it has x, y, and z.
The dot product (also called inner product) returns a scalar. If A·B = 0, the vectors are perpendicular. The sign tells you whether the angle between them is acute (positive) or obtuse (negative).
Only for 3D vectors. The result is a vector perpendicular to both inputs, with magnitude equal to |A||B|sin θ. Use the right-hand rule to find the direction.
A unit vector is a vector of length 1 pointing in the same direction as the original. Computed as A / |A|. The zero vector has no unit vector (division by zero).
cos θ = (A · B) / (|A| · |B|). Take the inverse cosine to get the angle. The calculator returns both degrees and radians and shows the cos θ value used.
No. This calculator covers 2D and 3D, the common cases for geometry, physics, and graphics. For higher-dimensional vector arithmetic, use a CAS or a programming language.
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