{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n$ be a positive integer and let $v = (v_1, \\dots, v_n)$ be a [[vector|0QN3D80AWP]] in $\\mathbb{R}^n$.\nThe length of $v$ is the real number\n$$\\|v\\| = \\sqrt{v_1^2 + v_2^2 + \\dots + v_n^2} = \\sqrt{v \\cdot v},$$\nwhere $v \\cdot v = v_1^2 + \\dots + v_n^2$ is the dot product of $v$ with itself.\nA unit vector is a vector of length 1.\n\n## Remarks\n\nFor each vector $v$, $\\|v\\| \\geq 0$, and $\\|v\\| = 0$ if and only if $v$ is the zero vector.\nFor each real number $c$, $\\|c v\\| = |c| \\, \\|v\\|$. Thus for $v \\neq 0$ the vector $v / \\|v\\|$ is a unit vector.\nOther names for the length are the magnitude and the Euclidean norm. The notation $|v|$ is also common.\n\n## Explanations\n\n### L1 human\n\nThe length of a vector is how far the vector moves you. In the plane, the vector $(3, 4)$ is the hypotenuse of a right triangle with legs 3 and 4. So its length is $\\sqrt{3^2 + 4^2} = 5$.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "defines": "0XKBKGTCPC",
    "kind": "definition",
    "msc": [
      "51N20"
    ],
    "name": "Length of a vector",
    "sources": [
      {
        "locator": "Length",
        "url": "https://en.wikipedia.org/wiki/Euclidean_vector"
      }
    ],
    "tag": "0DZS7ZG0QY",
    "uses": [
      "0QN3D80AWP"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0DZS7ZG0QY/",
  "links": [
    {
      "field": "defines",
      "target": "0XKBKGTCPC"
    },
    {
      "field": "uses",
      "target": "0QN3D80AWP"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0DZS7ZG0QY.md",
  "msc": [
    "51N20"
  ],
  "name": "Length of a vector",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0DZ/0DZS7ZG0QY.md",
  "status": null,
  "tag": "0DZS7ZG0QY",
  "uri": "m2://0DZS7ZG0QY",
  "url": "https://math2.tamnd.com/wiki/length-of-a-vector-definition/",
  "writing": {
    "tag": "0DZS7ZG0QY",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}