Question

If `veca, vecb, vecc` are three non-coplanar vectors, then the value of `(veca.(vecb xx vecc))/((vecc xx veca).vecb) + (vecb.(veca xx vecc))/(vecc.(veca xx vecb))` is ______.

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Answer 1

If `veca, vecb, vecc` are three non-coplanar vectors, then the value of `(veca.(vecb xx vecc))/((vecc xx veca).vecb) + (vecb.(veca xx vecc))/(vecc.(veca xx vecb))` is 0.

Explanation:

By definition of scalar triple product

`veca.(vecb xx vecc)` can be written as `[(veca, vecb, vecc)]`

`(veca.(vecb xx vecc))/((vecc xx veca).vecb) + (vecb.(veca xx vecc))/(vecc.(veca xx vecb)) = ([(veca, vecb, vecc)])/([(vecc, veca, vecb)]) + ([(vecb, veca, vecc)])/([(vecc, veca, vecb)])`

= `([(veca, vecb, vecc)])/([(veca, vecb, vecc)]) - ([(veca, vecb, vecc)])/([(veca, vecb, vecc)])` = 1 – 1 = 0

∵ `[(veca, vecb, vecc)] = [(vecb, vecc, veca)] = [(vecc, veca, vecb)]`

but `[(vecb, veca, vecc)] = -[(veca, vecb, vecc)]`