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Plücker coordinates
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=== Plane equations === If the point <math>\mathbf z = (z_0:z_1:z_2:z_3)</math> lies on {{mvar|L}}, then the columns of : <math> \begin{bmatrix} x_0 & y_0 & z_0 \\ x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3 \end{bmatrix} </math> are [[linearly dependent]], so that the rank of this larger matrix is still 2. This implies that all 3×3 submatrices have determinant zero, generating four (4 choose 3) plane equations, such as : <math> \begin{align} 0 & = \begin{vmatrix} x_0 & y_0 & z_0 \\ x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \end{vmatrix} \\[5pt] & = \begin{vmatrix} x_1 & y_1 \\ x_2 & y_2 \end{vmatrix} z_0 - \begin{vmatrix} x_0 & y_0 \\ x_2 & y_2 \end{vmatrix} z_1 + \begin{vmatrix} x_0 & y_0 \\ x_1 & y_1 \end{vmatrix} z_2 \\[5pt] & = p_{12} z_0 - p_{02} z_1 + p_{01} z_2 . \\[5pt] & = p^{03} z_0 + p^{13} z_1 + p^{23} z_2 . \end{align} </math> The four possible planes obtained are as follows. : <math> \begin{matrix} 0 & = & {}+ p_{12} z_0 & {}- p_{02} z_1 & {}+ p_{01} z_2 & \\ 0 & = & {}- p_{31} z_0 & {}- p_{03} z_1 & & {}+ p_{01} z_3 \\ 0 & = & {}+p_{23} z_0 & & {}- p_{03} z_2 & {}+ p_{02} z_3 \\ 0 & = & & {}+p_{23} z_1 & {}+ p_{31} z_2 & {}+ p_{12} z_3 \end{matrix} </math> Using dual coordinates, and letting {{math|(''a''{{sup|0}} : ''a''{{sup|1}} : ''a''{{sup|2}} : ''a''{{sup|3}})}} be the line coefficients, each of these is simply {{math|1=''a<sup>i</sup>'' = ''p<sup>ij</sup>''}}, or : <math> 0 = \sum_{i=0}^3 p^{ij} z_i , \qquad j = 0,\ldots,3 . </math> Each Plücker coordinate appears in two of the four equations, each time multiplying a different variable; and as at least one of the coordinates is nonzero, we are guaranteed non-vacuous equations for two distinct planes intersecting in {{mvar|L}}. Thus the Plücker coordinates of a line determine that line uniquely, and the map α is an [[Injective function|injection]].
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