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  "body": "\n## Definition\n\nLet $F$ be a [[field|04C5ZBG1WT]] and let $L$ be a line of the [[projective space|ZT9A6JK8QF]] $\\mathbb{P}^3(F)$.\nLet $x = (x_0, x_1, x_2, x_3)$ and $y = (y_0, y_1, y_2, y_3)$ be [[linearly independent|0CRMGPAH83]] vectors of $F^4$ with $L$ the line through $[x]$ and $[y]$.\nFor $0 \\le i \u003c j \\le 3$ put\n$$p_{ij} = x_i y_j - x_j y_i.$$\nThe *Plücker coordinates* of $L$ are the point\n$$[p_{01} : p_{02} : p_{03} : p_{12} : p_{13} : p_{23}] \\in \\mathbb{P}^5(F).$$\n\n## Remarks\n\nThe $p_{ij}$ are the $2 \\times 2$ minors of the matrix with rows $x$ and $y$.\nIf $x, y$ are replaced by another basis $x' = a x + b y$, $y' = c x + d y$ of the same plane, each $p_{ij}$ is multiplied by the [[determinant|00KDZRW4C0]] $ad - bc \\ne 0$, so the point of $\\mathbb{P}^5(F)$ depends only on $L$.\nThe coordinates satisfy the Plücker relation\n$$p_{01} p_{23} - p_{02} p_{13} + p_{03} p_{12} = 0.$$\nIn the language of the [[exterior algebra|0MDMZ965VE]], the $p_{ij}$ are the coordinates of $x \\wedge y$ in the basis $e_i \\wedge e_j$ of $\\Lambda^2 F^4$.\n\nExample: for $x = (1, 0, 0, 0)$ and $y = (0, 1, 0, 0)$, one gets $p_{01} = 1$ and all other $p_{ij} = 0$, so the line has Plücker coordinates $[1 : 0 : 0 : 0 : 0 : 0]$, and the relation holds since every product in it is $0$.\n\nNon-example: if $y = 2x$, every $p_{ij}$ is $0$, which is not a point of $\\mathbb{P}^5(F)$; the vectors must be linearly independent to span a line.\n",
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