How to Find Normal Vector?

For a line in 2D, if the direction vector is ( langle a, b rangle ), a normal vector is ( langle -b, a rangle ) or ( langle b, -a rangle )

For a plane in 3D written as (ax + by + cz = d), the normal vector is ( langle a, b, c rangle )

For a plane through three points, find two direction vectors in the plane and take their cross product

For a surface given by (F(x,y,z)=0), the normal vector at a point is ( nabla F = langle F_x, F_y, F_z rangle )

For a parametric surface ( mathbf{r}(u,v) ), compute ( mathbf{r}_u times mathbf{r}_v )

For a tangent line to a curve, a normal vector is any vector perpendicular to the tangent direction

To verify a normal vector, check that its dot product with the tangent direction is zero

Any nonzero scalar multiple of a normal vector is also a normal vector