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Question
If `vec a, vec b, vec c` are unit vectors such that `veca+vecb+vecc=0`, then write the value of `vec a.vecb+vecb.vecc+vecc.vec a`.
Solution
`veca, vecb, vecc` are unit examples.
`|veca| = |vecb| = |vecc| = 1`
`veca + vecb + vecc = 0`
`veca + vecb = -vecc` .....(i)
`veca xx (veca + vecb) = veca xx (-vecc)`
`veca xx veca + veca xx vecb = -veca xx vecc`
`|veca|^2 + veca xx vecb + vecc xx veca = 0`
`veca xx vecb + vecc xx veca = -1` ....(ii)
Again, from equation (i)
`vecb xx (veca + vecb) = vecb xx (-vecc)`
`vecb xx veca + vecb xx vecb = -vecb xx vecc`
`vecb xx vecc + veca xx vecb + |vecb|^2 = 0`
`veca xx vecb + vecb xx vecc = -1` ....(iii)
Again, from equation (i)
`vecc xx (veca + vecb) = vecc xx (-vecc)`
`vecc xx veca + vecc xx vecb = -vecc xx vecc`
`vecc xx veca + vecc xx vecb + |vecc|^2 = 0`
`vecc xx veca + vecc xx vecb = -1` ....(iv)
On adding equations (i) and (iv)
`2(veca xx vecb + vecb xx vecc + vecc xx veca) = -3`
`veca xx vecb + vecb xx vecc + vecc xx veca = -3/2`
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