Vector Calculator

Two Ways to Multiply

A vector is a quantity that has both a size and a direction — a velocity, a force, a displacement — and representing it as a list of components, like (3, 4) or (1, 2, 2), lets us calculate with it. Adding vectors places them tip to tail; scaling one stretches it; but the two operations that make vectors so powerful are the two ways of multiplying them. The dot product multiplies two vectors to give a single number that measures how much they point in the same direction: it is largest when they are parallel, zero when they are perpendicular, and negative when they oppose. From the dot product you get the angle between two vectors, the work done by a force, and the projection of one vector onto another — the "shadow" A casts along B. This calculator computes all of these, for vectors in two or three dimensions, the instant you type.

The cross product is the other multiplication, and it is special to three dimensions. It takes two vectors and produces a third vector at right angles to both, whose length equals the area of the parallelogram they span and whose direction follows the right-hand rule. The cross product is how you find a surface's normal direction in computer graphics, the torque a force produces about a pivot, and the magnetic force on a moving charge. In two dimensions there is no perpendicular direction left over, so the cross product collapses to a single signed number — the z-component — which still tells you the signed area and which way one vector turns relative to the other; this calculator reports that for 2D and the full vector for 3D. Alongside these it gives each vector's magnitude (its length, by the Pythagorean theorem), the unit vectors that capture pure direction, and the straight-line distance between the two vectors' tips. Everything is computed locally in your browser.

Vectors at Work

Vectors are the native language of physics and engineering, where almost every quantity that matters — force, velocity, acceleration, momentum, electric and magnetic fields — has a direction as well as a magnitude. The dot product computes work and energy; the cross product computes torque and angular momentum. In computer graphics and games, surface normals found by cross products determine how light reflects, and dot products decide how brightly a surface is lit. Navigation and robotics use vectors to represent position and heading; machine learning represents words, images and users as high-dimensional vectors and measures their similarity with — exactly — the dot product and the angle between them, the same quantities this tool computes. Even the projection shown here is the heart of least-squares fitting, the technique behind linear regression. Whether you are studying vector algebra, checking a physics problem, or building intuition for how direction and magnitude combine, this calculator gives you every standard vector quantity at once, instantly and privately.