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Question
For any vectors `bar"a", bar"b", bar"c"` show that `(bar"a" + bar"b" + bar"c") xx bar"c" + (bar"a" + bar"b" + bar"c") xx bar"b" + (bar"b" - bar"c") xx bar"a" = 2bar"a" xx bar"c"`
Solution
LHS = `(bar"a" + bar"b" + bar"c") xx bar"c" + (bar"a" + bar"b" + bar"c") xx bar"b" + (bar"b" - bar"c") xx bar"a"`
`= bar"a" xx bar"c" + bar"b" xx bar"c" + bar"c" xx bar"c" + bar"a" xx bar"b" + bar"b" xx bar"b" + bar"c" xx bar"b" + bar"b" xx bar"a" - bar"c" xx bar"a"`
`= bar"a" xx bar"c" + bar"b" xx bar"c" + bar0 + bar"a" xx bar"b" + bar"0" - bar"b" xx bar"c" - bar"a" xx bar"b" + bar"a" xx bar"c" ......[because bar"a" xx bar"b" = - bar"b" xx bar"a"]`
`= 2bar"a" xx bar"c"` = RHS.
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