Question

If the volume of parallelepiped formed by vectors $a\times b,b\times c$ and $c\times a$ is $36$ cubic units, then
$\begin{array}{ccc}& \text{List I}& \text{List II}\\ \text{(I)}& \text{Volume of parallelopiped formed by vectors}a,b\text{and}c\text{is}& \text{(P) 0 cubic units}\\ \text{(II)}& \text{Volume of tetrahedron formed by vectors}a,b\text{and}c\text{is}& \text{(Q) 12 cubic units}\\ \text{(III)}& \text{Volume of parallelopiped formed by vectors}a+b,b+c\text{and}c+a\text{is}& \text{(R) 6 cubic units}\\ \text{(IV)}& \text{Volume of parallelopiped formed by vectors}a-b,b-c\text{and}c-a\text{is}& \text{(S) 1 cubic units}\\ & & \text{(T) 5 cubic units}\end{array}$
Which of the following is CORRECT combination?

• A. $\text{(I)}\to \left(Q\right);\text{(II)}\to \left(P\right);\text{(III)}\to (R,T);\text{(IV)}\to \left(S\right)$
• B. $\text{(I)}\to \left(P\right);\text{(II)}\to (Q,T);\text{(III)}\to \left(R\right);\text{(IV)}\to \left(S\right)$
• C. $\text{(I)}\to \left(Q\right);\text{(II)}\to \left(P\right);\text{(III)}\to (S,T);\text{(IV)}\to \left(R\right)$
• D. $\text{(I)}\to \left(R\right);\text{(II)}\to \left(S\right);\text{(III)}\to \left(Q\right);\text{(IV)}\to \left(P\right)$

Answer 1

The correct option is D $\text{(I)}\to \left(R\right);\text{(II)}\to \left(S\right);\text{(III)}\to \left(Q\right);\text{(IV)}\to \left(P\right)$

Given, $[a\times bb\times cc\times a]=36$
$\left[abc\right]=6$
$\Rightarrow$ Volume of tetrahedron from by vectors
$a,b$ and $c$ is $\frac{1}{6}\left[abc\right]=1$
$[a+bb+cc+a]=2\left[abc\right]=12$
$a-b,b-c$ and $c-a$ are coplanar
$\Rightarrow$ $[a-bb-cc-a]=0$