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Question
For any three vectors `veca, vecb, vecc`, show that `veca - vecb, vecb - vecc, vecc - veca` are coplanar.
Solution
`[veca - vecb, vecb - vecc, vecc - veca]`
= `(veca -vecb). { (vecb - vecc) xx ( vecc - veca)}`
= `(veca- vecb). { vecb xx vecc - vecb xx veca - vecc xx vecc + vecc xx veca}`
= `(veca- vecb). { vecb xx vecc + veca xx vecb - veca xx vecc }`
= `veca . (vecb xx vecc) - veca . (veca xx vecb ) - a . (vec a xx vecc) - vecb . ( vecb xx vecc) - b .(veca xx vecb) + vecb. ( veca xx vecc) `
= ` veca vecb vecc - veca vecb vecc = 0`
Hence, `veca - vecb, vecb - vecc, vecc - veca` are coplanar.
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