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`.

Answer 1

`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`