Push a shopping cart at an angle and only part of your effort moves it forward — the rest presses uselessly sideways into the handle. The dot product measures exactly that: how much of one vector points along another. Multiply matching components and add: (1, 2, 3) · (4, 5, 6) = 4 + 10 + 18 = 32. The result is a single number, not a vector, and its sign tells a story — positive means the vectors roughly agree in direction, zero means they're perpendicular, negative means they oppose each other.

The geometric payoff is the angle formula: A · B = |A||B|cos θ, which turns the dot product into an angle-finder. Game developers use it to check whether an enemy is in front of the player (positive dot product with the facing direction) or behind; physicists use it for work, W = F · d, where only the force along the displacement counts. This calculator does the full job: enter the x, y, z components of Vector A and Vector B, and read off the dot product, both magnitudes, and the angle between them in radians and degrees. The worked example [1,2,3] · [4,5,6] and the perpendicular-vectors preset give you instant reference points for what "aligned" and "square" look like numerically.

With the pre-filled defaults — Vector A — x 1, Vector A — y 2, Vector A — z 3, Vector B — x 4, Vector B — y 5, Vector B — z 6 — the Dot Product Calculator returns Dot Product (A·B) 32.0000, |A| (magnitude of A) 3.7417, |B| (magnitude of B) 8.7750, Angle (radians) 0.225726, Angle (degrees) 12.9332. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.