Question

If `bar"a", bar"b", bar"c"` are three non-coplanar vectors, then `(bar"a" + bar"b" + bar"c").[(bar"a" + bar"b") xx (bar"a" + bar"c")] = - [bar"a"  bar"b" bar"c"]`

Answer 1

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

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

`= (bar"a" + bar"b" + bar"c")[bar"a" xx bar"a" + bar"a" xx bar"c" + bar"b" xx bar"a" + bar"b" xx bar"c"]`

`= bar"a".(bar"a" xx bar"c") + bar"a".(bar"b" xx bar"a") xx bar"a".(bar"b" xx bar"c") + bar"b".(bar"a" xx bar"c") + bar"b".(bar"b" xx bar"a") + bar"b".(bar"b" xx bar"c") + bar"c".(bar"a" xx bar"c") + bar"c".(bar"b" xx bar"a") xx bar"c".(bar"b" xx bar"c")     . .......[∵ bar"a" xx bar"a" = bar"0"]`

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

`= 0 + 0 + [bar"a" bar"b" bar"c"] - [bar"a" bar"b" bar"c"] + 0 + 0 + 0 - [bar"a" bar"b" bar"c"] + 0`

= `- [bar"a" bar"b" bar"c"]` = RHS.