Statement 1: Any vector in space can be uniquely written as the linear combination of three non-coplanar vectors.
Stetement 2: If `veca, vecb, vecc` are three non-coplanar vectors and `vecr` is any vector in space then
`[(veca,vecb, vecc)]vecc+[(vecb, vecc, vecr)]veca+[(vecc, veca, vecr)]vecb=[(veca, vecb, vecc)]vecr`

Options

A. 1
B. 2
C. 3
D. 4

Answer

B

Explanation

Clearly statemnet -1 is true
We have `vecr=xveca+yvecb+zvecc` ……………..i
Taking product successively with `vecbxxvecc, veccxxveca` and `vecaxxvecb`, we obtain
`x=([(vecb, vecc,vecr)])/([(veca, vecb, vecc)]),y=([(vecc, veca, vecr)])/([(veca, vecb, vecc)]),z=([(veca, vecb, vecr)])/([(veca, vecb, vecc)])`
Substituting the values of `x,y,z` in (i) we get
`[(vecb, vecc, vecr)]veca+[(vecc, veca, vecr)]vecb+[(veca, vecb, vecr)]vecr=[(veca, vecb, vecc)]vecr`
So Statement 2 is true. But, statement2 is not a correct explanation for statement -1