Vector Calculator (2D)
Enter the components of two 2D vectors to find their sum, difference, dot product, magnitudes, and the angle between them.
How 2D vector operations work
Vector addition and subtraction combine the x and y components separately: A + B = (Ax+Bx, Ay+By). The dot product, A·B = Ax×Bx + Ay×By, combines two vectors into a single number related to how aligned they are.
A vector's magnitude (length), |A| = √(Ax² + Ay²), comes from the Pythagorean theorem. The angle between two vectors is found from the dot product formula rearranged: cos(θ) = (A·B) / (|A| × |B|).
Frequently asked questions
What does a dot product of zero mean?
A dot product of zero means the two vectors are perpendicular (at a 90° angle) to each other, since cos(90°) = 0.
How is vector magnitude calculated?
Using the Pythagorean theorem: |A| = √(Ax² + Ay²) — the same formula used to find the hypotenuse of a right triangle with legs Ax and Ay.
What's the difference between a vector and a point?
A point represents a fixed location, while a vector represents a displacement or direction and magnitude — the same (x, y) numbers can represent either, depending on context, but vectors support operations like addition and dot products that points don't.