Question

Let the vectors $\overrightarrow{PQ},\overrightarrow{QR},\overrightarrow{RS},\overrightarrow{ST},\overrightarrow{TU}$ and $\overrightarrow{UP}$ represent the sides of a regular hexagon.
Statement - 1: $\overrightarrow{PQ}\times (\overrightarrow{RS}+\overrightarrow{ST})\ne 0$
Statement - 2: $\overrightarrow{PQ}\times \overrightarrow{RS}=0$ and $\overrightarrow{PQ}\times \overrightarrow{ST}\ne 0$

• A. Statement $-1$ is True, Statement $-2$ is True; Statement $-2$ is the correct explanation of Statement $-1$.
• B. Statement $-1$ is True, Statement $-2$ is True; Statement $-2$ is the incorrect explanation of Statement $-1$.
• C. Statement $-1$ is True, Statement $-2$ is False.
• D. Statement $-1$ is False, Statement $-2$ is True.

Answer 1

The correct option is C Statement $-1$ is True, Statement $-2$ is False.


We have $\overrightarrow{RS}+\overrightarrow{ST}=\overrightarrow{RT}.$ As $\overrightarrow{PQ}$ and $\overrightarrow{TR}$ are not parallel to each other,
$\Rightarrow \overrightarrow{PQ}\times (\overrightarrow{RS}+\overrightarrow{ST})=\overrightarrow{PQ}\times \overrightarrow{RT}\ne 0.$
Thus Statement -$1$ is true.
Since $\overrightarrow{PQ}$ and $\overrightarrow{RS}$ are not parallel, $\overrightarrow{PQ}\times \overrightarrow{RS}\ne 0.$
But $\overrightarrow{PQ}$ and $\overrightarrow{ST}$ are parallel to each other
$\Rightarrow \overrightarrow{PQ}\times \overrightarrow{ST}=0$
Therefore, the Statement -$2$ is false.