Editing Reflection (mathematics) (section)

==Reflection through a hyperplane in ''n'' dimensions==
Given a vector <math>v</math> in [[Euclidean space]] <math>\mathbb R^n</math>, the formula for the reflection in the [[hyperplane]] through the origin, [[orthogonal]] to <math>a</math>, is given by

:<math>\operatorname{Ref}_a(v) = v - 2\frac{v\cdot a}{a\cdot a}a,</math>

where <math>v\cdot a</math> denotes the [[dot product]] of <math>v</math> with <math>a</math>. Note that the second term in the above equation is just twice the [[vector projection]] of <math>v</math> onto <math>a</math>. One can easily check that
*{{math|1=Ref<sub>''a''</sub>(''v'') = −''v''}}, if <math>v</math> is parallel to <math>a</math>, and
*{{math|1=Ref<sub>''a''</sub>(''v'') =  ''v''}}, if <math>v</math> is perpendicular to {{mvar|''a''}}.

Using the [[geometric product]], the formula is

:<math>\operatorname{Ref}_a(v) = -\frac{a v a}{a^2} .</math>

Since these reflections are isometries of Euclidean space fixing the origin they may be represented by [[orthogonal matrices]]. The orthogonal matrix corresponding to the above reflection is the [[Matrix (mathematics)|matrix]]

:<math>R = I-2\frac{aa^T}{a^Ta},</math>

where <math>I</math> denotes the <math>n \times n</math> [[identity matrix]] and <math>a^T</math> is the [[transpose]] of a. Its entries are

:<math>R_{ij} = \delta_{ij} - 2\frac{a_i a_j}{ \left\| a \right\| ^2 },</math>

where {{math|''δ''<sub>''ij''</sub>}} is the [[Kronecker delta]].

The formula for the reflection in the affine hyperplane <math>v\cdot a=c</math> not through the origin is
:<math>\operatorname{Ref}_{a,c}(v) = v - 2\frac{v \cdot a - c}{a\cdot a}a.</math>