Det finns två sorters trippelprodukter av vektorer ; den skalära och den vektoriella. Båda handlar om att multiplicera tre vektorer (
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{\displaystyle {\textbf {a}},{\textbf {b}},{\textbf {c}}}
skalär- och kryssprodukter .
Den skalära trippelprodukten definieras som skalärprodukten av den ena vektorn med kryssprodukten av de två andra, dvs:
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{\displaystyle {\textbf {a}}\cdot ({\textbf {b}}\times {\textbf {c}})}
Vektorerna kan inom produkten flyttas runt cykliskt, dvs:
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{\displaystyle {\textbf {a}}\cdot ({\textbf {b}}\times {\textbf {c}})={\textbf {b}}\cdot ({\textbf {c}}\times {\textbf {a}})={\textbf {c}}\cdot ({\textbf {a}}\times {\textbf {b}})}
Den skalära trippelprodukten kan geometriskt tolkas som volymen (med tecken) av parallellepipeden som definieras av de tre vektorerna.
Man kan också tolka den skalära trippelprodukten som determinanten av den matris som har de tre vektorerna som rader eller kolonner.
Den vektoriella trippelprodukten är
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{\displaystyle {\textbf {a}}\times ({\textbf {b}}\times {\textbf {c}})}
Den vektoriella trippelprodukten kan utvecklas med hjälp av Lagranges formel [ 1]
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{\displaystyle {\textbf {a}}\times ({\textbf {b}}\times {\textbf {c}})={\textbf {b}}({\textbf {a}}\cdot {\textbf {c}})-{\textbf {c}}({\textbf {a}}\cdot {\textbf {b}})}
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{\displaystyle {\textbf {d}}=(d_{x},d_{y},d_{z})={\textbf {a}}\times ({\textbf {b}}\times {\textbf {c}})=(a_{x},a_{y},a_{z})\times ((b_{x},b_{y},b_{z})\times (c_{x},c_{y},c_{z}))=}
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{\displaystyle =(a_{x},a_{y},a_{z})\times (b_{y}c_{z}-b_{z}c_{y},\ b_{z}c_{x}-b_{x}c_{z},\ b_{x}c_{y}-b_{y}c_{x})}
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{\displaystyle d_{x}=a_{y}b_{x}c_{y}-a_{y}b_{y}c_{x}-a_{z}b_{z}c_{x}+a_{z}b_{x}c_{z}}
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{\displaystyle d_{y}=a_{z}b_{y}c_{z}-a_{z}b_{z}c_{y}-a_{x}b_{x}c_{y}+a_{x}b_{y}c_{x}}
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{\displaystyle d_{z}=a_{x}b_{z}c_{x}-a_{x}b_{x}c_{z}-a_{y}b_{y}c_{z}+a_{y}b_{z}c_{y}}
Utveckling av
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{\displaystyle d_{x}}
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{\displaystyle d_{x}=a_{y}b_{x}c_{y}-a_{y}b_{y}c_{x}-a_{z}b_{z}c_{x}+a_{z}b_{x}c_{z}=}
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{\displaystyle =a_{y}b_{x}c_{y}-a_{y}b_{y}c_{x}-a_{z}b_{z}c_{x}+a_{z}b_{x}c_{z}+a_{x}b_{x}c_{x}-a_{x}b_{x}c_{x}=}
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{\displaystyle =b_{x}(a_{y}c_{y}+a_{z}c_{z}+a_{x}c_{x})-c_{x}(a_{y}b_{y}+a_{z}b_{z}+a_{x}b_{x})=}
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{\displaystyle =b_{x}({\textbf {a}}\cdot {\textbf {c}})-c_{x}({\textbf {a}}\cdot {\textbf {b}})}
På samma sätt får vi:
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{\displaystyle d_{y}=b_{y}({\textbf {a}}\cdot {\textbf {c}})-c_{y}({\textbf {a}}\cdot {\textbf {b}})}
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{\displaystyle d_{z}=b_{z}({\textbf {a}}\cdot {\textbf {c}})-c_{z}({\textbf {a}}\cdot {\textbf {b}})}
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{\displaystyle {\textbf {d}}=(d_{x},d_{y},d_{z})=(b_{x}({\textbf {a}}\cdot {\textbf {c}})-c_{x}({\textbf {a}}\cdot {\textbf {b}}),\ b_{y}({\textbf {a}}\cdot {\textbf {c}})-c_{y}({\textbf {a}}\cdot {\textbf {b}}),\ b_{z}({\textbf {a}}\cdot {\textbf {c}})-c_{z}({\textbf {a}}\cdot {\textbf {b}}))=}
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{\displaystyle =(b_{x}({\textbf {a}}\cdot {\textbf {c}}),\ b_{y}({\textbf {a}}\cdot {\textbf {c}}),\ b_{z}({\textbf {a}}\cdot {\textbf {c}}))-(c_{x}({\textbf {a}}\cdot {\textbf {b}}),\ c_{y}({\textbf {a}}\cdot {\textbf {b}}),\ c_{z}({\textbf {a}}\cdot {\textbf {b}}))=}
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{\displaystyle =(b_{x},\ b_{y},\ b_{z})({\textbf {a}}\cdot {\textbf {c}})-(c_{x},\ c_{y},\ c_{z})({\textbf {a}}\cdot {\textbf {b}})}
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{\displaystyle ={\textbf {b}}({\textbf {a}}\cdot {\textbf {c}})-{\textbf {c}}({\textbf {a}}\cdot {\textbf {b}})}
^ Uppkallad efter Joseph-Louis Lagrange som använde metoden i komponentform 1773 i Solutions analytiques de quelques problèmes sur les pyramides triangulaires William Rowan Hamilton introducerade begreppen "vektor" och "skalär" 1843).
