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  "body": "\n## Summary\n\nPlücker coordinates describe a line of three-dimensional projective space by six homogeneous numbers, the $2 \\times 2$ minors of a matrix whose rows span the line.\nThey turn the set of lines of $\\mathbb{P}^3$ into the points of a quadric in $\\mathbb{P}^5$, the [[Klein quadric|6WV71T5CK1]].\nMore generally, the same minors embed every [[Grassmannian|CYPA882RB9]] into a projective space.\n\n## Explanations\n\n### L1 human\n\nA point of projective space has coordinates, but a line is a bigger object, so how does one give it coordinates?\nPick two points on the line and write their coordinates as the two rows of a $2 \\times 4$ matrix.\nThe six $2 \\times 2$ minors of that matrix do not depend, up to a common factor, on which two points were picked.\nThese six numbers are the Plücker coordinates of the line.\nFor the line through $[1 : 0 : 0 : 0]$ and $[0 : 1 : 0 : 0]$, only one minor is nonzero.\n",
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