Vector Calculator

Vectors encode immutable physical quantities containing both magnitude and angular direction across spatial dimensions. The dot product quantitatively evaluates parallel alignment efficiency, while the cross product generates a strictly orthogonal resultant, forming the mathematical backbone of 3D electrodynamics and mechanics.

This tool runs the two ways vectors multiply. The dot product returns a single number measuring how much two vectors point the same way — zero means they are perpendicular. The cross product (3D) returns a whole new vector at right angles to both, whose length equals the area of the parallelogram they span. To find the angle, just rearrange the dot product: cos θ = (a·b)/(|a| |b|).

The dot product works in any number of dimensions, but a true cross product — two vectors in, one vector perpendicular to both out — exists only in three and seven dimensions. This oddity ties directly to the algebras of quaternions and octonions, the only normed division algebras beyond the real and complex numbers. So the familiar 3D cross product is far less universal than it looks.