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Question
Prove that `[vec"a" - vec"b", vec"b" - vec"c", vec"c" - vec"a"]` = 0
Solution
`[vec"a" - vec"b", vec"b" - vec"c", vec"c" - vec"a"] = (vec"a" - vec"b")*[(vec"b" - vec"c") xx (vec"c" - vec"a")]`
= `(vec"a" - vec"b")*[vec"b" xx vec"c" - vec"b" xx vec"a" - vec"c" xx vec"c" + vec"c" xx vec"a"]`
= `vec"a"*(vec"b" xx vec"c") - vec"a"*(vec"b" xx vec"a") - vec"a"*(vec"c" xx vec"c") + vec"a"*(vec"c" xx vec"a") - vec"b"*(vec"b" xx vec"c") + vec"b"*(vec"b" xx vec"a") + vec"b"*(vec"c" xx vec"c") - vec"b"*(vec"c" xx vec"a")`
= `vec"a"*(vec"b" xx vec"c") - 0 - 0 + 0 - 0 + 0 + 0 - vec"b"*(vec"c" xx vec"a")`
= `[vec"a", vec"b", vec"c"] - [vec"a", vec"b", vec"c"]` = 0
Hence proved.
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