Question

If $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ be three non-zero and non-coplanar vectors and $\overrightarrow{p},\overrightarrow{q}$ and $\overrightarrow{r}$ be three vectors given by $\overrightarrow{p}=\overrightarrow{a}+\overrightarrow{b}-2\overrightarrow{c},\overrightarrow{q}=3\overrightarrow{a}-2\overrightarrow{b}+\overrightarrow{c},$ and $\overrightarrow{r}=\overrightarrow{a}-4\overrightarrow{b}+2\overrightarrow{c}.$ If the volume of the parallelepiped determined by $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ is $V_1$and that of the parallelopiped determined by $\overrightarrow{p},\overrightarrow{q}$ and $\overrightarrow{r}$ is $V_2$ then $V_2:V_1$ is

• A. 1 : 15
• B. 15 : 1
• C. 4 : 5
• D. 5 : 4

Answer 1

Answer: (B)

We have $\vec{p}=\vec{a}+\vec{b}-2\vec{c}$
$\vec{q}=3\vec{a}-2\vec{b}+\vec{c}$
$\vec{r}=\vec{a}-4\vec{b}+2\vec{c}$
Also $V_1=\left[\vec{a}\vec{b}\vec{c}\right]$ and $V_2=\left[\vec{p}\vec{q}\vec{r}\right]$
$\therefore V_2=\left|\begin{array}{ccc}1& 1& -2\\ 3& -2& 1\\ 1& -4& 2\end{array}\right|\left[\vec{a}\vec{b}\vec{c}\right]$
$=\left|\begin{array}{ccc}1& 1& -2\\ 0& -5& 7\\ 0& -5& 4\end{array}\right|\left[\vec{a}\vec{b}\vec{c}\right]$
$=15\left[\vec{a}\vec{b}\vec{c}\right]=15V_1$
$\therefore V_2:V_1=15:1$