Triple scalar product: Difference between revisions
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<div class="definition"><div class="short_definition">The [[scalar]] '''A''' · ('''B''' × '''C''') written ('''ABC''') or ['''ABC'''], where '''A''', '''B''', and '''C''' are any three vectors.</div><br/> <div class="paragraph">The dot denotes a [[scalar product]] and the cross a [[vector product]]. When '''A''', '''B''', and '''C''' are written in terms of their components along the ''x'', ''y'', and ''z'' axes of the [[rectangular Cartesian coordinates]], that is, '''A''' = ''a''<sub>1</sub>'''i''' + ''a''<sub>2</sub>'''j''' + ''a''<sub>3</sub>'''k''', '''B''' = ''b''<sub>1</sub>'''i''' + ''b''<sub>2</sub>'''j''' + ''b''<sub>3</sub>'''k''', and '''C''' = ''c''<sub>1</sub>'''i''' + ''c''<sub>2</sub>'''j''' + ''c''<sub>3</sub>'''k''', the triple scalar product is the determinant <div class="display-formula"><blockquote>[[File:ams2001glos-Te43.gif|link=|center|ams2001glos-Te43]]</blockquote></div> Any cyclic change among the vectors in a triple product does not alter its value.</div><br/> </div> | <div class="definition"><div class="short_definition">The [[scalar]] '''A''' · ('''B''' × '''C''') written ('''ABC''') or ['''ABC'''], where '''A''', '''B''', and '''C''' are any three vectors.</div><br/> <div class="paragraph">The dot denotes a [[scalar product]] and the cross a [[vector product]]. When '''A''', '''B''', and '''C''' are written in terms of their components along the ''x'', ''y'', and ''z'' axes of the [[rectangular Cartesian coordinates|rectangular Cartesian coordinates]], that is, '''A''' = ''a''<sub>1</sub>'''i''' + ''a''<sub>2</sub>'''j''' + ''a''<sub>3</sub>'''k''', '''B''' = ''b''<sub>1</sub>'''i''' + ''b''<sub>2</sub>'''j''' + ''b''<sub>3</sub>'''k''', and '''C''' = ''c''<sub>1</sub>'''i''' + ''c''<sub>2</sub>'''j''' + ''c''<sub>3</sub>'''k''', the triple scalar product is the determinant <div class="display-formula"><blockquote>[[File:ams2001glos-Te43.gif|link=|center|ams2001glos-Te43]]</blockquote></div> Any cyclic change among the vectors in a triple product does not alter its value.</div><br/> </div> | ||
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Latest revision as of 17:08, 25 April 2012
triple scalar product
The scalar A · (B × C) written (ABC) or [ABC], where A, B, and C are any three vectors.
The dot denotes a scalar product and the cross a vector product. When A, B, and C are written in terms of their components along the x, y, and z axes of the rectangular Cartesian coordinates, that is, A = a1i + a2j + a3k, B = b1i + b2j + b3k, and C = c1i + c2j + c3k, the triple scalar product is the determinant Any cyclic change among the vectors in a triple product does not alter its value.
