How this calculator works
Divide every component by the vector's magnitude to preserve direction while giving length one. The planar direction is the counterclockwise angle from positive x in the xy plane. In three dimensions it is the azimuth, while three direction angles describe alignment with the positive axes. A vector on the z axis has no planar azimuth, so that output is omitted. A missing unit component is determined only up to sign.
Formula
u = v/|v|; |v| = sqrt(x^2 + y^2 + z^2). cos(alpha_i) = u_i. A missing component is plus or minus sqrt(1 - sum of known components squared).
Worked example
(3, 4) has magnitude 5 and unit vector (0.6, 0.8). Its planar direction is about 53.1301 degrees. If a planar unit vector has known x = 0.6, its other component can be either 0.8 or -0.8.
Advertisement
Frequently asked questions
Is direction the same as magnitude?
No. Scaling a vector by a positive factor changes its magnitude while leaving its direction unchanged. A negative scale reverses direction.
Why are two missing components returned?
Squaring loses the sign. Both values have the required unit length unless the missing component is zero, when they coincide.
Can the zero vector be normalized?
No. Its magnitude is zero, so division cannot produce a unique unit direction.