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Prove that abcababcabc(a¯+2b¯-c¯).[(a¯-b¯)×(a¯-b¯-c¯)]=3[a¯b¯c¯]. - Mathematics and Statistics

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Sum

Prove that `(bar"a" + 2bar"b" - bar"c"). [(bar"a" - bar"b") xx (bar"a" - bar"b" - bar"c")] = 3 [bar"a" bar"b" bar"c"]`.

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Solution

`(bar"a" + 2bar"b" - bar"c"). [(bar"a" - bar"b") xx (bar"a" - bar"b" - bar"c")] = 3 [bar"a" bar"b" bar"c"]`

`= (bar"a" + 2bar"b" - bar"c").(bar"a"xxbar"a" - bar"a" xx bar"b" - bar"a" xx bar"c" - bar"b" xx bar"a" + bar"b" xx bar"b" + bar"b" xx bar"c")`

`= (bar"a" + 2bar"b" - bar"c").(bar"0" - bar"a" xx bar"b" - bar"c" xx bar"a" - bar"a" xx bar"b" + bar"0" + bar"b" xx bar"c")`

`= (bar"a" + 2bar"b" - bar"c"). (bar"c" xx bar"a" + bar"b" xx bar"c")`

`= bar"a".(bar"c" xx bar"a") + bar"a".(bar"b"xxbar"c") + 2bar"b".(bar"c" xx bar"a") + 2bar"b".(bar"b"xxbar"c") - bar"c".(bar"c" xx bar"a") - bar"c".(bar"b" xx bar"c")`

`= 0 + bar"a".(bar"b" xx bar"c") + 2bar"b".(bar"c" xx bar"a") + 2 xx 0 - 0 - 0`

`= [bar"a"  bar"b"  bar"c"] + 2[bar"b"  bar"c"  bar"a"]`

`= [bar"a"  bar"b"  bar"c"] + 2[bar"a"  bar"b"  bar"c"] = 3[bar"a"  bar"b"  bar"c"]`

Concept: Vector Triple Product
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