{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "03NK6RSRQ1"
    }
  ],
  "body": "\n## Statement\n\nLet $n$ be a positive integer, and let $u$ and $v$ be nonzero vectors in $\\mathbb{R}^n$.\nLet $u \\cdot v = u_1 v_1 + \\dots + u_n v_n$ be their dot product, let $\\|u\\|$ and $\\|v\\|$ be their [[lengths|0XKBKGTCPC]], and let $\\theta$ be the [[angle between them|0ZEHZY85E6]].\nThen\n$$u \\cdot v = \\|u\\| \\, \\|v\\| \\cos \\theta.$$\n\n## Remarks\n\nWhen the angle is defined by its cosine, the formula follows from the definition. The content is that this angle is the angle of elementary geometry. In the plane, write $u = \\|u\\| (\\cos \\alpha, \\sin \\alpha)$ and $v = \\|v\\| (\\cos \\beta, \\sin \\beta)$. Then $u \\cdot v = \\|u\\| \\, \\|v\\| \\cos(\\alpha - \\beta)$ by the addition formula for the cosine.\nThe formula also follows from the law of cosines in the triangle with the vertices $0$, $u$ and $v$.\nIf $u = 0$ or $v = 0$, then both sides are $0$ for each value of $\\theta$.\n\n## Explanations\n\n### L1 human\n\nThe dot product is large and positive when two vectors point in nearly the same direction. It is zero when they are perpendicular, and negative when they point in nearly opposite directions. For unit vectors $u$ and $v$, the dot product is just the cosine of the angle between them.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "51N20"
    ],
    "name": "Geometric formula for the dot product",
    "role": "proposition",
    "sources": [
      {
        "locator": "Equivalence of the definitions",
        "url": "https://en.wikipedia.org/wiki/Dot_product"
      }
    ],
    "tag": "070TXS9AXM",
    "uses": [
      "0ZEHZY85E6",
      "0XKBKGTCPC"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/070TXS9AXM/",
  "links": [
    {
      "field": "uses",
      "target": "0ZEHZY85E6"
    },
    {
      "field": "uses",
      "target": "0XKBKGTCPC"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/070TXS9AXM.md",
  "msc": [
    "51N20"
  ],
  "name": "Geometric formula for the dot product",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/070/070TXS9AXM.md",
  "status": null,
  "tag": "070TXS9AXM",
  "uri": "m2://070TXS9AXM",
  "url": "https://math2.tamnd.com/wiki/geometric-formula-for-the-dot-product/",
  "writing": {
    "tag": "070TXS9AXM",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}