Dot Product of Two 3D Vectors (Components)

Adding a third axis costs the dot product exactly one more term. Its most-used property survives untouched: a·b = 0 means the two vectors meet at a right angle. That test underpins an enormous amount of working code — checking whether a surface normal faces a light, whether a ray runs parallel to a plane, whether two axes of a rotation matrix have drifted out of square. Example: (1, 2, 2)·(2, −2, 1) = 2 − 4 + 2 = 0, so those two are perpendicular even though neither lies along an axis and no picture would make it obvious.

Solve for one missing component and the equation is just a linear one in disguise: with a = (1, 2, 2), b_x = 2, b_y = −2 and a required dot product of 0, the remaining component must satisfy 2 − 4 + 2b_z = 0, giving b_z = 1. The paired coefficient has to be non-zero, though — if a_z = 0 then b_z never enters the sum, and no dot product can reveal it.