TheToolNet
Straightforward tools. Useful answers. No account required.

Orthogonal Vector Calculator

Enter a nonzero 2D or 3D vector to generate one orthogonal vector, its unit direction, and a dot-product verification.

Written and checked by TheToolNet Editorial Team · Last reviewed: September 20, 2026 · Editorial method

Find an orthogonal vector

Enter 2 or 3 components separated by commas or spaces.

How the construction works

For a 2D vector ⟨x,y⟩, ⟨−y,x⟩ is perpendicular because x(−y)+y(x)=0. In 3D there is an entire plane of orthogonal directions; the calculator crosses the input with a stable coordinate axis to return one example.

Orthogonal does not mean unique

Any nonzero scalar multiple remains orthogonal. Use the dot-product calculator to compare two chosen vectors, or the unit-vector calculator to normalize a direction.