Statement If `G_(1),G_(2),G_(3)` are the centroids of the triangular faces `OBC, OCA, OAB` of a tetrahedron `OABC`, then the ratio of the volume of the tetrahedron to that of the parallelopiped with `OG_(1),OG_(2),OG_(3)` as coterminous edges is `9:4`.
Statement 2: For any three vctors,`veca, vecb,vecc`
`[(veca+vecb, vecb+vecc, vecc+veca)]=2[(veca, vecb, vecc)]`

Options

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

Answer

A

Explanation

Clearly statemnet 2 is true.
With reference to the orign `O`,let the position vectors of he vertices `A,B,C` be `veca, vecb` and `vecc` respectively. The, the position vectors of `G_(1),G_(2),G_(3)` are `(vecb+vecc)/3,(vecc+veca)/3,(veca+vecb)/3` respectively. Let `V_(1)=1/6[(veca,vecb,vecc)]`
Let `V_(2)` be the volume of the parallelopiped with `OG_(1),OG_(2),OG_(3)` as coterminus edges. Then,
`V_(2)=[(vecOG_(1), vecOG_(2),vecOG_(3))]`
`impliesV_(2)=[((vecb+vecc)/3 (vecc+veca)/3 (veca+vecb)/3)]`
`impliesV_(2)=1/27[(vecb+vecc,vecc+veca,veca+vecb)]=2/27[(veca,vecb,vecc)]`
`:.(V_(1))/(V_(2))=(1/6[(veca,vecb,vecc)])/(2/27[(veca,vecb,vecc)])=9/4`
Hence `V_(1):V_(2)=9:4`
So statement 1 is true.
Also, statement-2 is a correct explanation of statement -1.