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Question
Show that the vectors `veca, vecb` are coplanar if `veca+vecb, vecb+vecc ` are coplanar.
Solution
It is given that `veca+vecb, vecb+vecc `
Therefore
Scalar triple product=Volume of the parallelopoid=0
`(veca+vecb).[(vecb+vecc)xx(vecc+veca)]=0`
`=>(veca+vecb).|__(vecbxxvecc)+(vecbxxveca)+0+(veccxxveca)__|=0`
`=>veca.(vecbxxvecc)+veca.(vecbxxveca)+veca.(veccxxveca)+vecb.(vecbxxvecc)+vecb.(veccxxveca)=0`
⇒ [abc]+0+0+0+0+[bca]=0
⇒2[abc]=0
⇒[abc]=0
Therefore, the vectors `veca,vecb `are coplanar.
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