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Show that abcbcacab0a¯×(b¯×c¯)+b¯×(c¯×a¯)+c¯×(a¯×b¯)=0¯ - Mathematics and Statistics

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Sum

Show that `bar"a" xx (bar"b" xx bar"c") + bar"b" xx (bar"c" xx bar"a") + bar"c" xx (bar"a" xx bar"b") = bar"0"`

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Solution

LHS = `bar"a" xx (bar"b" xx bar"c") + bar"b" xx (bar"c" xx bar"a") + bar"c" xx (bar"a" xx bar"b")`

`= (bar"a".bar"c")bar"b" - (bar"a".bar"b")bar"c" + (bar"b".bar"a")bar"c" - (bar"b".bar"c")bar"a" + (bar"c".bar"b")bar"a" - (bar"c".bar"a")bar"b"`

`= (bar"c".bar"a")bar"b" - (bar"a".bar"b")bar"c" + (bar"a".bar"b")bar"c" - (bar"b".bar"c")bar"a" + (bar"b".bar"c")bar"a" - (bar"c".bar"a")bar"b"`  .....[∵ bar"a".bar"b" = bar"b".bar"a"]

= `bar"0"` = RHS  

Concept: Vector Triple Product
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