Editing Normal (geometry) (section)

==Hypersurfaces in ''n''-dimensional space==
For an <math>(n-1)</math>-dimensional [[hyperplane]] in [[n-dimensional space|<math>n</math>-dimensional space]] <math>\R^n</math> given by its parametric representation
<math display=block>\mathbf{r}\left(t_1, \ldots, t_{n-1}\right) = \mathbf{p}_0 + t_1 \mathbf{v}_1 + \cdots + t_{n-1}\mathbf{v}_{n-1},</math>
where <math>\mathbf{p}_0</math> is a point on the hyperplane and <math>\mathbf{v}_i</math> for <math>i = 1, \ldots, n - 1</math> are linearly independent vectors pointing along the hyperplane, a normal to the hyperplane is any vector <math>\mathbf n</math> in the [[null space]] of  the matrix <math>V = \begin{bmatrix}\mathbf{v}_1 & \cdots &\mathbf{v}_{n-1}\end{bmatrix},</math> meaning {{tmath|1=V\mathbf n = \mathbf 0}}. That is, any vector orthogonal to all in-plane vectors is by definition a surface normal. Alternatively, if the hyperplane is defined as the solution set of a single linear equation {{tmath|1=a_1x_1+\cdots+a_nx_n = c}}, then the vector <math>\mathbf{n} = \left(a_1, \ldots, a_n\right)</math> is a normal.

The definition of a normal to a surface in three-dimensional space can be extended to <math>(n - 1)</math>-dimensional [[hypersurface]]s in {{tmath|\R^n}}. A hypersurface may be [[Local property|locally]] defined implicitly as the set of points <math>(x_1, x_2, \ldots, x_n)</math> satisfying an equation {{tmath|1=F(x_1, x_2, \ldots, x_n) = 0}}, where <math>F</math> is a given [[Scalar field|scalar function]]. If <math>F</math> is [[continuously differentiable]] then the hypersurface is a [[differentiable manifold]] in the [[Neighbourhood (mathematics)|neighbourhood]] of the points where the [[gradient]] is not zero. At these points a normal vector is given by the gradient:
<math display=block>\mathbb n = \nabla F\left(x_1, x_2, \ldots, x_n\right) = \left( \tfrac{\partial F}{\partial x_1}, \tfrac{\partial F}{\partial x_2}, \ldots, \tfrac{\partial F}{\partial x_n} \right)\,.</math>

The '''normal line''' is the one-dimensional subspace with basis <math>\{\mathbf{n}\}.</math>

A vector that is normal to the space spanned by the linearly independent vectors {{math|'''v'''{{sub|1}}, ..., '''v'''{{sub|''r''−1}}}} and falls within the {{nowrap|{{mvar|r}}-dimensional}} space spanned by the linearly independent vectors  {{math|'''v'''{{sub|1}}, ..., '''v'''{{sub|''r''}}}} is given by the {{nowrap|{{mvar|r}}-th}} column of the matrix {{math|1=Λ = ''V''(''V''{{isup|T}}''V''){{sup|−1}}}}, where the matrix {{math|1=''V'' = ('''v'''{{sub|1}}, ..., '''v'''{{sub|''r''}})}} is the juxtaposition of the {{mvar|r}} column vectors.  (Proof: {{math|1=''V''{{isup|T}}Λ = ''I''}} so each of {{math|'''v'''{{sub|1}}, ..., '''v'''{{sub|''r''−1}}}} is perpendicular to the last column of {{math|Λ}}.)  This formula works even when {{mvar|r}} is less than the dimension of the Euclidean {{nowrap|space  {{mvar|n}}.}}  The formula simplifies to {{math|1=Λ = (''V''{{isup|T}}){{sup|−1}}}} when {{math|1=''r'' = ''n''}}.