{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $a$ and $b$ be vectors in $\\mathbb{R}^3$, and let $\\times$ be the [[cross product|0N5N15HWZ7]]. Then\n$$b \\times a = -(a \\times b).$$\n\n## Remarks\n\nWith $b = a$ the statement gives $a \\times a = -(a \\times a)$, thus $a \\times a = 0$ for each vector $a$.\nThe proof is a direct computation: each component of $b \\times a$ is the component of $a \\times b$ with the two products in the other order.\nThe cross product is thus not commutative. For example, $e_1 \\times e_2 = e_3$ but $e_2 \\times e_1 = -e_3$.\n\n## Explanations\n\n### L1 human\n\nIf you swap the two factors of a cross product, the result points the other way. Its length stays the same. The right-hand rule shows this: swapping the factors turns your hand over.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "15A72"
    ],
    "name": "Anticommutativity of the cross product",
    "role": "proposition",
    "sources": [
      {
        "locator": "Algebraic properties",
        "url": "https://en.wikipedia.org/wiki/Cross_product"
      }
    ],
    "tag": "07ZS685JJC",
    "uses": [
      "0N5N15HWZ7"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/07ZS685JJC/",
  "links": [
    {
      "field": "uses",
      "target": "0N5N15HWZ7"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/07ZS685JJC.md",
  "msc": [
    "15A72"
  ],
  "name": "Anticommutativity of the cross product",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/07Z/07ZS685JJC.md",
  "status": null,
  "tag": "07ZS685JJC",
  "uri": "m2://07ZS685JJC",
  "url": "https://math2.tamnd.com/wiki/anticommutativity-of-the-cross-product/",
  "writing": {
    "tag": "07ZS685JJC",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}