Editing Plücker coordinates (section)

=== Line-line crossing ===
{{further|Line-line intersection}}

Two lines in {{tmath|\mathbb P^3}} are either [[skew lines|skew]] or [[coplanar]], and in the latter case they are either coincident or intersect in a unique point. If {{mvar|p<sub>ij</sub>}} and {{mvar|p&prime;<sub>ij</sub>}} are the Plücker coordinates of two lines, then they are coplanar precisely when

:<math>\mathbf d \cdot \mathbf m' + \mathbf m \cdot \mathbf d' = 0,</math>

as shown by

: <math>
\begin{align}
0 & = p_{01}p'_{23} + p_{02}p'_{31} + p_{03}p'_{12} + p_{23}p'_{01} + p_{31}p'_{02} + p_{12}p'_{03} \\[5pt]
& = \begin{vmatrix}x_0&y_0&x'_0&y'_0\\
x_1&y_1&x'_1&y'_1\\
x_2&y_2&x'_2&y'_2\\
x_3&y_3&x'_3&y'_3\end{vmatrix}.
\end{align}
</math>

When the lines are skew, the sign of the result indicates the sense of crossing: positive if a right-handed screw takes {{mvar|L}} into {{mvar|L&prime;}}, else negative.

The quadratic Plücker relation essentially states that a line is coplanar with itself.