Normal Vectors From Cross Products

From Tangent Directions to a Normal

A parametrized surface gives a point \(\mathbf r(u,v)\) for each pair of parameters in a domain. Changing \(u\) while holding \(v\) fixed, or changing \(v\) while holding \(u\) fixed, traces curves on the surface. Their tangent vectors are the partial derivatives \(\mathbf r_u\) and \(\mathbf r_v\).

At a regular point, these two tangent vectors are linearly independent. Their cross product is perpendicular to both, so it points in a direction normal to the surface. This construction is useful because it turns the surface’s parametrization directly into a normal vector, without first finding an equation for the surface.

A normal vector need not have length one. Every nonzero scalar multiple of a normal is also perpendicular to the same tangent directions. If a unit normal is needed, divide a nonzero normal vector by its length. The cross product itself is often preferable in calculations because it retains useful information about the scale of the tangent vectors.

Computing the Cross Product

Write the parametrization in components as \(\mathbf r(u,v)=(x(u,v),y(u,v),z(u,v))\). First calculate both partial derivatives, then apply the component formula for the cross product:

The determinant notation is a compact way to remember the component formula. In practical calculations, expanding each component explicitly helps prevent sign errors. The result is a vector depending on \(u\) and \(v\); evaluating it at a particular parameter pair gives a normal vector at the corresponding surface point, provided the result is nonzero.

Why the Cross Product Is Perpendicular

The perpendicularity is not an additional assumption about surfaces. It is an algebraic property of the cross product. For vectors \(\mathbf a=(a_1,a_2,a_3)\) and \(\mathbf b=(b_1,b_2,b_3)\), its component formula is

Taking a dot product with \(\mathbf a\) and expanding gives pairs of equal terms with opposite signs:

The same expansion with \(\mathbf b\) gives

Thus \(\mathbf a\times\mathbf b\) is perpendicular to both vectors. Taking \(\mathbf a=\mathbf r_u\) and \(\mathbf b=\mathbf r_v\) proves that the cross product of the surface’s tangent vectors is perpendicular to both coordinate tangent directions. At a regular point those directions span the tangent plane, so the cross product is normal to that plane.

Normals to a Sphere Patch

For a sphere of radius \(R>0\) centered at the origin, one useful parametrization is

The parameter \(u\) controls latitude and \(v\) controls position around the vertical axis. Differentiating component by component gives

Normals to a Cylinder Patch

A cylinder of radius \(a>0\) around the \(z\)-axis can be parametrized by

Its partial derivatives are

Common Mistake: A Sign Error in the Middle Component

A frequent error is to change the sign of the middle component when expanding the determinant. For example, in the plane calculation above, an incorrect expansion might give

This cannot be the correct cross product because its dot product with the first tangent vector is

The middle component must be \(z_u x_v-x_u z_v\), not its negative. Substituting the plane’s tangent vectors gives \(1\cdot1-2\cdot2=-3\), so the corrected cross product is \((-3,-3,3)\). Its dot products with the two tangent vectors are both zero, as required.

When This Construction Applies

The procedure is the same for any differentiable parametrization: compute \(\mathbf r_u\), compute \(\mathbf r_v\), take their cross product, and check that it is nonzero. When it is nonzero, it supplies a normal to the tangent plane at the corresponding point. The plane, sphere, and cylinder examples illustrate different geometries, but none requires a different rule.

If the cross product vanishes, do not report the zero vector as a normal. A normal vector must be nonzero, and the vanishing cross product says that the two parameter tangent vectors do not span a tangent plane at that parameter pair. The parametrization may be singular there, or a different local parametrization may be needed to describe the surface regularly.

For a reliable calculation, keep the parameter dependence until after differentiating, write the cross product components in order, and use dot products as a final check. A correct nonzero result must have zero dot product with each of the two partial derivative vectors.