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 $V_{2}=\left[\vec{p}\,\vec{q}\,\vec{r}\right]$
$\therefore V_{2}=\begin{vmatrix}1&1&-2\\ 3&-2&1\\ 1&-4&2\end{vmatrix}\left[\vec{a}\,\vec{b}\,\vec{c}\right]$
$=\begin{vmatrix}1&1&-2\\ 0&-5&7\\ 0&-5&4\end{vmatrix}\left[\vec{a}\,\vec{b}\,\vec{c}\right]$
$=15\left[\vec{a}\,\vec{b}\,\vec{c}\right]=15\,V_{1}$
$\therefore V_{2} : V_{1}=15 : 1$