{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $u = (u_1, u_2)$ and $v = (v_1, v_2)$ be nonzero vectors in the [[Euclidean plane|0XW7V9ZZ05]] $\\mathbb{R}^2$. Let $u \\cdot v = u_1 v_1 + u_2 v_2$ and $\\det(u, v) = u_1 v_2 - u_2 v_1$.\nThe oriented angle from $u$ to $v$ is the class modulo $2\\pi$ of each real number $\\theta$ with\n$$\\cos \\theta = \\frac{u \\cdot v}{\\|u\\| \\, \\|v\\|} \\quad \\text{and} \\quad \\sin \\theta = \\frac{\\det(u, v)}{\\|u\\| \\, \\|v\\|}.$$\nFor points $A$, $B$ and $C$ of $\\mathbb{R}^2$ with $A \\neq B$ and $C \\neq B$, the oriented angle $\\measuredangle ABC$ is the oriented angle from $A - B$ to $C - B$.\n\n## Remarks\n\nSuch a $\\theta$ exists because $(u \\cdot v)^2 + \\det(u, v)^2 = \\|u\\|^2 \\|v\\|^2$. It is unique modulo $2\\pi$.\nSwapping the vectors changes the sign: the oriented angle from $v$ to $u$ is minus the oriented angle from $u$ to $v$.\nThe [[unoriented angle|0ZEHZY85E6]] between $u$ and $v$ is the absolute value of the representative of the oriented angle in $(-\\pi, \\pi]$.\nMathlib's `EuclideanGeometry.oangle` works in each oriented Euclidean plane, with values in $\\mathbb{R} / 2\\pi\\mathbb{Z}$. It gives $0$ when a vector is zero.\n\n## Explanations\n\n### L1 human\n\nThe oriented angle remembers the direction of the turn. Turning from $(1, 0)$ to $(0, 1)$ is a quarter turn anticlockwise, so the oriented angle is $\\pi / 2$. Turning back, from $(0, 1)$ to $(1, 0)$, gives $-\\pi / 2$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "03YN70NK0Z",
    "kind": "definition",
    "msc": [
      "51M05"
    ],
    "name": "Oriented angle",
    "sources": [
      {
        "locator": "Signed angles",
        "url": "https://en.wikipedia.org/wiki/Angle"
      }
    ],
    "tag": "0G4W89YJRK",
    "uses": [
      "0XW7V9ZZ05",
      "0ZEHZY85E6"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0G4W89YJRK/",
  "links": [
    {
      "field": "defines",
      "target": "03YN70NK0Z"
    },
    {
      "field": "uses",
      "target": "0XW7V9ZZ05"
    },
    {
      "field": "uses",
      "target": "0ZEHZY85E6"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0G4W89YJRK.md",
  "msc": [
    "51M05"
  ],
  "name": "Oriented angle",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0G4/0G4W89YJRK.md",
  "status": null,
  "tag": "0G4W89YJRK",
  "uri": "m2://0G4W89YJRK",
  "url": "https://math2.tamnd.com/wiki/oriented-angle-definition/",
  "writing": {
    "tag": "0G4W89YJRK",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}