➡️Vector Calculator

About this tool

Enter two vectors with two or three components each, such as 1, 2, 3, plus a scalar if you need it. The calculator lists a + b, a − b, the dot and cross products, both magnitudes and unit vectors, the angle between them, the projection of a onto b, the scalar multiple and the area of the parallelogram they span. It also tells you when the vectors are perpendicular or parallel.

How vector operations work

A vector has both a size and a direction, written as components such as (1, 2, 3). Adding and subtracting work component by component. The magnitude, or length, is the square root of the sum of the squared components: |(1, 2, 3)| = √14 ≈ 3.742. Dividing a vector by its magnitude gives a unit vector pointing the same way. The dot product, a₁b₁ + a₂b₂ + a₃b₃, is a single number. It's zero when the vectors are perpendicular and links to the angle between them: cos θ = a · b ÷ (|a||b|). The cross product of two 3D vectors is a new vector perpendicular to both, following the right-hand rule, and its length is the area of the parallelogram they form. For a = (1, 2, 3) and b = (4, 5, 6), a · b = 32, a × b = (−3, 6, −3) and the angle between them is about 12.93°. The projection of a onto b is the shadow of a along b's direction. These operations underpin physics, where work is a dot product and torque a cross product, and 3D graphics, where cross products give surface normals. Honest limits: the cross product is defined for 3D vectors; 2D inputs are treated as having z = 0, so only a z-component appears. Results are rounded to six decimal places. The matrix calculator handles transformations and the scientific calculator covers trigonometry.

Frequently asked questions