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 kan tolkas som volymen av en parallelepiped. Basytan är i figuren
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{\displaystyle {\textbf {base}}={\textbf {b}}\times {\textbf {c}}=||{\textbf {b}}||\cdot ||{\textbf {c}}||\sin \theta }
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{\displaystyle h=||{\textbf {a}}||\cdot \cos \alpha }
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{\displaystyle ({\textbf {b}}\times {\textbf {c}})\cdot {\textbf {a}}={\textbf {a}}\cdot ({\textbf {b}}\times {\textbf {c}})}
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{\displaystyle {\textbf {b}}}
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{\displaystyle \theta }
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{\displaystyle \sin(-\theta )=-\sin \theta }
Den skalära trippelprodukten definieras som skalärprodukten av den ena vektorn med kryssprodukten av de två andra, det vill säga:
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{\displaystyle {\textbf {a}}\cdot ({\textbf {b}}\times {\textbf {c}})=({\textbf {b}}\times {\textbf {c}})\cdot {\textbf {a}}}
Vektorerna kan inom produkten flyttas runt cykliskt , det vill säga:
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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}})}
Byter man ordning i kryssprodukten byter trippelprodukten tecken:
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{\displaystyle {\textbf {a}}\cdot ({\textbf {b}}\times {\textbf {c}})=-{\textbf {a}}\cdot ({\textbf {c}}\times {\textbf {b}})}
Eftersom skalärprudukten är kommutativ gäller även exempelvis:
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{\displaystyle {\textbf {a}}\cdot ({\textbf {b}}\times {\textbf {c}})=({\textbf {a}}\times {\textbf {b}})\cdot {\textbf {c}}}
[ 1] 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 [ 2]
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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}})}
