{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $A$ be an [[affine space|09XNAPTJQ7]] over a field $k$, with space of translations $V$.\nA set $S$ of points of $A$ is *coplanar* if the vectors $q - p$, for $p, q \\in S$, span a [[linear subspace|00YC1D652K]] of [[dimension|03GSZ8RTZJ]] at most $2$.\n\n## Remarks\n\nWhen $A$ has dimension at least $2$, the set $S$ is coplanar exactly when some affine plane of $A$ contains $S$.\n\nExample: in $\\R^3$ the points $(0,0,0)$, $(1,0,0)$, $(0,1,0)$ and $(1,1,0)$ are coplanar.\nCheck: all differences lie in the subspace $\\set{z = 0}$, which has dimension $2$.\n\nNon-example: the points $(0,0,0)$, $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ of $\\R^3$ are not coplanar.\nThe differences from the origin are the three standard basis vectors, which span a space of dimension $3$.\n\nIn $\\R^3$, four points $p_0, p_1, p_2, p_3$ are coplanar exactly when the [[determinant|00KDZRW4C0]] of the $3 \\times 3$ matrix with columns $p_1 - p_0$, $p_2 - p_0$, $p_3 - p_0$ is zero.\nEvery set of at most three points is coplanar.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "4TWY53TBKK",
    "kind": "definition",
    "msc": [
      "51N10",
      "51A05"
    ],
    "name": "Coplanarity",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Coplanarity"
      }
    ],
    "tag": "TQJGPS6VZK",
    "uses": [
      "09XNAPTJQ7",
      "00YC1D652K",
      "03GSZ8RTZJ",
      "00KDZRW4C0"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/TQJGPS6VZK/",
  "links": [
    {
      "field": "defines",
      "target": "4TWY53TBKK"
    },
    {
      "field": "uses",
      "target": "09XNAPTJQ7"
    },
    {
      "field": "uses",
      "target": "00YC1D652K"
    },
    {
      "field": "uses",
      "target": "03GSZ8RTZJ"
    },
    {
      "field": "uses",
      "target": "00KDZRW4C0"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/TQJGPS6VZK.md",
  "msc": [
    "51N10",
    "51A05"
  ],
  "name": "Coplanarity",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TQJ/TQJGPS6VZK.md",
  "status": null,
  "tag": "TQJGPS6VZK",
  "uri": "m2://TQJGPS6VZK",
  "url": "https://tamnd.github.io/math2/wiki/coplanarity-definition/",
  "writing": {
    "tag": "TQJGPS6VZK",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}