Dot Product Calculator

This calculator finds the dot product of two three-dimensional vectors, along with the angle between them and the length of each, the core measurements you need when working with vectors. The dot product, also called the scalar product, multiplies two vectors to give a single number rather than another vector, and that number carries a lot of meaning. It measures how much the two vectors point in the same direction: it is largest when they are parallel, zero when they are perpendicular, and negative when they point in opposing directions. This makes the dot product indispensable across physics, engineering, computer graphics and machine learning, where it is used to calculate the work done by a force, to project one vector onto another, to test whether vectors are at right angles, and to measure similarity between data. This tool computes it. You enter the x, y and z components of each vector, and the calculator returns the dot product, the angle between the two vectors in degrees, and the magnitude, or length, of each vector. The results update as you type, so you can see how the angle shrinks as the vectors align and the dot product grows. Use it for physics and maths problems, for graphics and geometry, or to check vector homework. The relationship at its heart is elegant: the dot product equals the product of the two magnitudes times the cosine of the angle between them, which is exactly how the angle is recovered. A dot product of zero is the quickest test that two vectors are perpendicular. To work in two dimensions, simply set the z components to zero. The calculations are exact for your inputs, with the angle rounded for display.

a . b = ax bx + ay by + az bz. The dot product is zero when vectors are perpendicular. Set z to 0 for 2D. Angle rounded for display.