Editing Plücker coordinates (section)

=== Dual coordinates ===
Alternatively, a line can be described as the intersection of two planes. Let {{mvar|L}} 
be a line contained in distinct planes {{math|'''a'''}} and {{math|'''b'''}} with homogeneous coefficients {{math|(''a''{{sup|0}} : ''a''{{sup|1}} : ''a''{{sup|2}} : ''a''{{sup|3}})}} and {{math|(''b''{{sup|0}} : ''b''{{sup|1}} : ''b''{{sup|2}} : ''b''{{sup|3}})}}, respectively. (The first plane equation is <math display=inline>\sum_k a^k x_k =0,</math> for example.) The dual Plücker coordinate {{mvar|p<sup>ij</sup>}} is

:<math>p^{ij} = \begin{vmatrix} a^{i} & a^{j} \\ b^{i} & b^{j}\end{vmatrix}  = a^{i}b^{j}-a^{j}b^{i} . </math>

Dual coordinates are convenient in some computations, and they are equivalent to primary coordinates:
: <math>
(p_{01}:p_{02}:p_{03}:p_{23}:p_{31}:p_{12})=
(p^{23}:p^{31}:p^{12}:p^{01}:p^{02}:p^{03})
</math>
Here, equality between the two vectors in homogeneous coordinates means that the numbers on the right side are equal to the numbers on the left side up to some common scaling factor {{math|λ}}. Specifically, let {{math|(''i'', ''j'', ''k'', ''ℓ'')}} be an [[even permutation]] of {{math|(0, 1, 2, 3)}}; then

: <math>p_{ij} = \lambda p^{k\ell} .  </math>