Det finns två sorters trippelprodukter av vektorer ; den skalära och den vektoriella. Båda handlar om att multiplicera tre vektorer (a , b , c {\displaystyle {\textbf {a}},{\textbf {b}},{\textbf {c}}} ) med varandra genom en serie skalär- och kryssprodukter .
Skalär trippelprodukt
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Vektoriell trippelprodukt
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Den vektoriella trippelprodukten är
a × ( b × c ) {\displaystyle {\textbf {a}}\times ({\textbf {b}}\times {\textbf {c}})}
Den vektoriella trippelprodukten kan utvecklas med hjälp av Lagranges formel [ 2] , "BAC-CAB-regeln":
a × ( b × c ) = b ( a ⋅ c ) − c ( a ⋅ b ) {\displaystyle {\textbf {a}}\times ({\textbf {b}}\times {\textbf {c}})={\textbf {b}}({\textbf {a}}\cdot {\textbf {c}})-{\textbf {c}}({\textbf {a}}\cdot {\textbf {b}})} Bevis
d = ( d x , d y , d z ) = a × ( b × c ) = ( a x , a y , a z ) × ( ( b x , b y , b z ) × ( c x , c y , c z ) ) = {\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}))=}
= ( a x , a y , a z ) × ( b y c z − b z c y , b z c x − b x c z , b x c y − b y c x ) {\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})} gerd 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 {\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}} ,
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 {\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}} och
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 {\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 d x {\displaystyle d_{x}} ger: 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 = {\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}=}
= 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 = {\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}=}
= 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 ) = {\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})=}
= b x ( a ⋅ c ) − c x ( a ⋅ b ) {\displaystyle =b_{x}({\textbf {a}}\cdot {\textbf {c}})-c_{x}({\textbf {a}}\cdot {\textbf {b}})} På samma sätt får vi:
d y = b y ( a ⋅ c ) − c y ( a ⋅ b ) {\displaystyle d_{y}=b_{y}({\textbf {a}}\cdot {\textbf {c}})-c_{y}({\textbf {a}}\cdot {\textbf {b}})} och
d z = b z ( a ⋅ c ) − c z ( a ⋅ b ) {\displaystyle d_{z}=b_{z}({\textbf {a}}\cdot {\textbf {c}})-c_{z}({\textbf {a}}\cdot {\textbf {b}})} , sålunda:
d = ( d x , d y , d z ) = ( b x ( a ⋅ c ) − c x ( a ⋅ b ) , b y ( a ⋅ c ) − c y ( a ⋅ b ) , b z ( a ⋅ c ) − c z ( a ⋅ b ) ) = {\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}}))=}
= ( b x ( a ⋅ c ) , b y ( a ⋅ c ) , b z ( a ⋅ c ) ) − ( c x ( a ⋅ b ) , c y ( a ⋅ b ) , c z ( a ⋅ b ) ) = {\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}}))=}
= ( b x , b y , b z ) ( a ⋅ c ) − ( c x , c y , c z ) ( a ⋅ b ) {\displaystyle =(b_{x},\ b_{y},\ b_{z})({\textbf {a}}\cdot {\textbf {c}})-(c_{x},\ c_{y},\ c_{z})({\textbf {a}}\cdot {\textbf {b}})}
= b ( a ⋅ c ) − c ( a ⋅ b ) {\displaystyle ={\textbf {b}}({\textbf {a}}\cdot {\textbf {c}})-{\textbf {c}}({\textbf {a}}\cdot {\textbf {b}})} Referenser och noter
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