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Four-dimensional space
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==Vectors== Mathematically, a four-dimensional space is a [[space (mathematics)|space]] that needs four parameters to specify a [[point (geometry)|point]] in it. For example, a general point might have position [[Euclidean vector|vector]] {{math|'''a'''}}, equal to : <math>\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \\ a_4 \end{pmatrix}.</math> This can be written in terms of the four [[standard basis]] vectors {{math|('''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub>, '''e'''<sub>4</sub>)}}, given by :<math>\mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}; \mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}; \mathbf{e}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}; \mathbf{e}_4 = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}, </math> so the general vector {{math|'''a'''}} is : <math> \mathbf{a} = a_1\mathbf{e}_1 + a_2\mathbf{e}_2 + a_3\mathbf{e}_3 + a_4\mathbf{e}_4.</math> Vectors add, subtract and scale as in three dimensions. The [[dot product]] of Euclidean three-dimensional space generalizes to four dimensions as : <math>\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 + a_4 b_4.</math> It can be used to calculate the [[norm (mathematics)|norm]] or [[Euclidean distance|length]] of a vector, :<math> \left| \mathbf{a} \right| = \sqrt{\mathbf{a} \cdot \mathbf{a} } = \sqrt{a_1^2 + a_2^2 + a_3^2 + a_4^2},</math> and calculate or define the [[angle]] between two non-zero vectors as :<math> \theta = \arccos{\frac{\mathbf{a} \cdot \mathbf{b}}{\left|\mathbf{a}\right| \left|\mathbf{b}\right|}}.</math> Minkowski spacetime is four-dimensional space with geometry defined by a non-degenerate [[pairing]] different from the dot product: : <math>\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 - a_4 b_4.</math> As an example, the distance squared between the points {{math|(0,0,0,0)}} and {{math|(1,1,1,0)}} is 3 in both the Euclidean and Minkowskian 4-spaces, while the distance squared between {{math|(0,0,0,0)}} and {{math|(1,1,1,1)}} is 4 in Euclidean space and 2 in Minkowski space; increasing {{math|''b''{{sub|4}}}} decreases the metric distance. This leads to many of the well-known apparent "paradoxes" of relativity. The [[cross product]] is not defined in four dimensions. Instead, the [[exterior product]] is used for some applications, and is defined as follows: : <math> \begin{align} \mathbf{a} \wedge \mathbf{b} = (a_1b_2 - a_2b_1)\mathbf{e}_{12} + (a_1b_3 - a_3b_1)\mathbf{e}_{13} + (a_1b_4 - a_4b_1)\mathbf{e}_{14} + (a_2b_3 - a_3b_2)\mathbf{e}_{23} \\ + (a_2b_4 - a_4b_2)\mathbf{e}_{24} + (a_3b_4 - a_4b_3)\mathbf{e}_{34}. \end{align}</math> This is [[bivector]] valued, with bivectors in four dimensions forming a [[six-dimensional space|six-dimensional]] linear space with basis {{math|('''e'''<sub>12</sub>, '''e'''<sub>13</sub>, '''e'''<sub>14</sub>, '''e'''<sub>23</sub>, '''e'''<sub>24</sub>, '''e'''<sub>34</sub>)}}. They can be used to generate rotations in four dimensions.
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