MATH · VECTOR
Vector Calculator
Calculate dot products, cross products, magnitudes, angles, and vector addition or subtraction for 2D and 3D vectors.
Dot Product A · B
11
SummaryA · B = 11
About This Calculator
Compute dot products, cross products, magnitudes, angles, and arithmetic operations on 2D and 3D vectors. Enter the components of vector A (and vector B where needed), choose an operation, and get the result instantly.
How It Works
Enter the x, y (and optionally z) components for each vector, then select the operation. The dot product returns a scalar; the cross product (3D only) returns a perpendicular vector; add/subtract return a new vector; magnitude returns the length; angle returns the angle in degrees between the two vectors.
The Formula
A·B = AxBx + AyBy + AzBz |A| = √(Ax² + Ay² + Az²) θ = arccos(A·B / (|A| × |B|))
- A, B
- input vectors
- A·B
- dot product (scalar)
- |A|
- magnitude (length) of A
- θ
- angle between A and B in degrees
Frequently Asked Questions
- What is the difference between a dot product and a cross product?
- The dot product produces a scalar (a single number) equal to the product of the magnitudes times the cosine of the angle between them. It measures how much two vectors point in the same direction. The cross product (3D only) produces a new vector perpendicular to both input vectors; its magnitude equals the area of the parallelogram they span.
- When is the dot product zero?
- The dot product is zero when the two vectors are perpendicular (orthogonal) to each other — the angle between them is 90°. This is a quick test for perpendicularity.
- Why is cross product only defined for 3D vectors?
- The cross product is defined in three dimensions because it requires a third axis for the resulting perpendicular vector. In 2D, you can compute the scalar "z-component" of the cross product (sometimes called the perp-dot product), but the full cross product lives in 3D.
- What does the magnitude of a vector represent?
- The magnitude (length) of a vector represents its size or scale — how far the vector extends in space. For a velocity vector, the magnitude is the speed; for a force vector, it is the force intensity.