Processing math: 100%
mail  mail@stemandmusic.in
    
call  +91-9818088802

Determinant Product in Arbitrary Non Standard Basis

  1. Just like Cross Product, the Determinant Product of Vectors can be calculated for Vectors given in Any Arbitrary Non Standard Basis.
  2. The following demonstrates the calculation of Determinant Product for 3 4-Dimensional Vectors →A, →B and →C represented in any Arbitrary Non Standard Basis →e1, →e2, →e3 and →e4

    →A=A1→e1+A2→e2+A3→e3+A4→e4→B=B1→e1+B2→e2+B3→e3+B4→e4→C=C1→e1+C2→e2+C3→e3+C4→e4

    [→A→B→C]D=|→e1→e2→e3→e4A1A2A3A4B1B2B3B4C1C2C3C4|

    ⇒[→A→B→C]D=|A2A3A4B2B3B4C2C3C4|[→e2→e3→e4]D−|A1A3A4B1B3B4C1C3C4|[→e1→e3→e4]D+|A1A2A4B1B2B4C1C2C4|[→e3→e4→e1]D−|A1A2A3B1B2B3C1C2C3|[→e1→e2→e3]D

    Setting →e5=[→e2→e3→e4]D,   →e6=[→e1→e3→e4]D,   →e7=[→e1→e2→e4]D   and   →e8=[→e1→e2→e3]D   we get

    [→A→B→C]D=|A2A3A4B2B3B4C2C3C4|→e5−|A1A3A4B1B3B4C1C3C4|→e6+|A1A2A4B1B2B4C1C2C4|→e7−|A1A2A3B1B2B3C1C2C3|→e8

    The Determinant Product in Non Standard Basis of N−1 Vectors of N-Dimensions (where N≥3) can be calculated similarly.
  3. Following examples demonstrates the calculation of the Determinant Product of Vectors →A, →B and →C represented in Non Standard Basis →e1, →e2, →e3 and →e4 as give below

    →e1=[1−234]→e2=[−3245]→e3=[5732]→e4=[−5143]

    →A=3→e1−2→e2+1→e3+3→e4

    →B=−3→e1+3→e2−2→e3+2→e4

    →C=1→e1+5→e2+2→e3−2→e4

    [→A→B→C]D=|→e1→e2→e3→e43−213−33−22152−2|

    ⇒[→A→B→C]D=|−2133−2252−2|[→e2→e3→e4]D−|313−3−2212−2|[→e1→e3→e4]D+|3−23−33215−2|[→e3→e4→e1]D−|3−21−33−2152|[→e1→e2→e3]D

    Calculating the Determinants and setting →e5=[→e2→e3→e4]D,   →e6=[→e1→e3→e4]D,   →e7=[→e1→e2→e4]D   and   →e8=[→e1→e2→e3]D   we get

    ⇒[→A→B→C]D=64→e5+16→e6−94→e7−22→e8

    The Basis Vectors →e5, →e6, →e7 and →e8 are calculated as follows

    →e5=[→e2→e3→e4]D=|^e1^e2^e3^e4−32455732−5143|

    ⇒→e5=|245732143|^e1−|−345532−543|^e2+|−325572−513|^e3−|−324573−514|^e4

    ⇒→e5=51^e1−72^e2+93^e3−15^e4=[51−7293−15]

    →e6=[→e1→e3→e4]D=|^e1^e2^e3^e41−2345732−5143|

    ⇒→e6=|−234732143|^e1−|134532−543|^e2+|1−24572−513|^e3−|1−23573−514|^e4

    ⇒→e6=41^e1−66^e2+229^e3−215^e4=[41−66229−215]

    →e7=[→e1→e2→e4]D=|^e1^e2^e3^e41−234−3245−5143|

    ⇒→e7=|−234245143|^e1−|134−345−543|^e2+|1−24−325−513|^e3−|1−23−324−514|^e4

    ⇒→e7=29^e1+24^e2+61^e3−41^e4=[292461−41]

    →e8=[→e1→e2→e3]D=|^e1^e2^e3^e41−234−32455732|

    ⇒→e8=|−234245732|^e1−|134−345532|^e2+|1−24−325572|^e3−|1−23−324573|^e4

    ⇒→e8=19^e1+30^e2−217^e3+173^e4=[1930−217173]

    Here ^e1, ^e2, ^e3 and ^e4 are Identity Orthonormal Basis Vectors.
Related Topics
Determinant Product of Vectors,    Introduction to Vector Algebra
  Home  |  TOC  |  Calculators
© Invincible IDeAS. All Rights Reserved