Vector Calculator

The dot product returns a scalar that measures alignment, with 0 for perpendicular nonzero vectors. This calculator supports the ordinary 3D cross product, which returns a vector perpendicular to both inputs and whose length equals the area of their parallelogram.

This calculator implements the ordinary vector cross product in 3D. For 2D vectors, the determinant ax * by - ay * bx gives a signed scalar area. You can also embed both vectors in 3D with z = 0, then read that determinant from the z component of the cross product.

The calculator uses the formula angle = arccos((a . b) / (|a| x |b|)), then converts radians to degrees. The result is always in the range 0-180 degrees. If either vector has zero magnitude, the angle is undefined and the calculator returns no result.

A unit vector has magnitude 1 and points in the same direction as the original vector. The calculator divides each component by the vector's magnitude to produce it. Use unit vectors when only direction matters, for example surface normals in 3D graphics or force directions in physics. The zero vector has no direction, so the calculator returns no result for it.

No, addition requires both vectors to have the same number of components. If you have a 2D vector and want to work alongside a 3D one, switch the calculator to 3D mode and enter z = 0 for the 2D vector; components are then added pair by pair.