Statement 1: Let `vecr` be any vector in space. Then, `vecr=(vecr.hati)hati+(vecr.hatj)hatj+(vecr.hatk)hatk`
Statement 2: If `veca, vecb, vecc` are three non-coplanar vectors and `vecr` is any vector in space then
`vecr={([(vecr, vecb, vecc)])/([(veca, vecb, vecc)])}veca+{([(vecr, vecc, veca)])/([(veca, vecb, vecc)])}vecb+{([(vecr, veca, vecb)])/([(veca, vecb, vecc)])}vecc`

Options

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

Answer

A

Explanation

Since `veca,vecb, vecc` are non coplanar vectors.
Therefore, there exists scalars x,y,z such that
`vecr=xveca+yvecb+zvecc`……………..i
Taking dot products with `vecbxxvecc, veccxxveca` and `vecaxxvecb` successively, we get
`vecr.(vecbxxvecc)=(xveca+yvecb+zvecc).(vecbxxvecc)`
`vecr.(veccxxveca)=(xvecaxxyvecb+zvecc).(veccxxveca)`
`vecr.(vecaxxvecb)=(xveca+yvecb+zvecc).(vecaxxvecb)`
`implies[(vecr, vecb, vecc)]=x[(veca, vecb, vecc)]`
`=[(vecr, vecc, veca)]=y[(vecb, vecc, veca)]`
and `[(vecr, veca, vecb)]=z[(vecc, veca, vecb)]`
`impliesx=([(vecr, vecb, vecc)])/([(veca, vecb, vecc)]),y=([(vecy, vecc, veca)])/([(veca, vecb, vecc)])` and `z=([(vecr, veca, vecb)])/([(veca, vecb, vecc)])`
Substituting the alues of `x,y,z` in (i) we get
`vecr={([(vecr, vecb, vecc)])/([(veca, vecb, vec)])}veca+{([(vecr, vecc, veca)])/([(veca, vecb, vecc)])}vecb+{([(vecr, veca, vecb)])/([(veca,vecb, vecc)])}vecc`
So, statement 2 is true.
On replacing `veca, vecb,` and `vecc` by `hati, hatj` and `hatk` respectively in statement -2 we obtain statement-1.
So, statement 1 is true and statement -2 is a correct explanation for statement -1