"exterior product" meaning in English

{
  "forms": [
    {
      "form": "exterior products",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "exterior product (plural exterior products)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Algebra",
          "orig": "en:Algebra",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "text": "The exterior product of two vectors is a bivector."
        },
        {
          "text": "The exterior product between two vectors is anti-commutative; therefore the exterior product between a vector and itself is zero."
        },
        {
          "text": "The exterior product between a multivector of grade k and a multivector of grade n is a multivector of grade k+n, unless k+n is larger than the dimension of the vector space to which the vectors belong (out of which the multivectors are constructed), in which case their product is zero."
        },
        {
          "text": "The exterior product between a multivector of grade k and a multivector of grade n is commutative if k times n is even and anti-commutative if k times n is odd. (This is related to what is meant when it is said that the exterior product is alternating. It means that a permutation of the factors of a wedge product of vectors changes the sign of the product if and only if the permutation is odd.)"
        }
      ],
      "glosses": [
        "A kind of product between vectors and/or multivectors which is associative, linear, and alternating."
      ],
      "id": "en-exterior_product-en-noun-FfRHFWQl",
      "links": [
        [
          "algebra",
          "algebra"
        ]
      ],
      "raw_glosses": [
        "(algebra) A kind of product between vectors and/or multivectors which is associative, linear, and alternating."
      ],
      "related": [
        {
          "word": "exterior algebra"
        },
        {
          "word": "exterior derivative"
        },
        {
          "word": "wedge product"
        }
      ],
      "synonyms": [
        {
          "word": "Grassmann product"
        }
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
    }
  ],
  "word": "exterior product"
}

{
  "forms": [
    {
      "form": "exterior products",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "exterior product (plural exterior products)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "exterior algebra"
    },
    {
      "word": "exterior derivative"
    },
    {
      "word": "wedge product"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "en:Algebra"
      ],
      "examples": [
        {
          "text": "The exterior product of two vectors is a bivector."
        },
        {
          "text": "The exterior product between two vectors is anti-commutative; therefore the exterior product between a vector and itself is zero."
        },
        {
          "text": "The exterior product between a multivector of grade k and a multivector of grade n is a multivector of grade k+n, unless k+n is larger than the dimension of the vector space to which the vectors belong (out of which the multivectors are constructed), in which case their product is zero."
        },
        {
          "text": "The exterior product between a multivector of grade k and a multivector of grade n is commutative if k times n is even and anti-commutative if k times n is odd. (This is related to what is meant when it is said that the exterior product is alternating. It means that a permutation of the factors of a wedge product of vectors changes the sign of the product if and only if the permutation is odd.)"
        }
      ],
      "glosses": [
        "A kind of product between vectors and/or multivectors which is associative, linear, and alternating."
      ],
      "links": [
        [
          "algebra",
          "algebra"
        ]
      ],
      "raw_glosses": [
        "(algebra) A kind of product between vectors and/or multivectors which is associative, linear, and alternating."
      ],
      "topics": [
        "algebra",
        "mathematics",
        "sciences"
      ]
    }
  ],
  "synonyms": [
    {
      "word": "Grassmann product"
    }
  ],
  "word": "exterior product"
}