Vectors 2 Question 14
14. Let the vectors $\overrightarrow{\mathbf{P Q}}, \overrightarrow{\mathbf{Q R}}, \overrightarrow{\mathbf{R S}}, \overrightarrow{\mathbf{S T}}, \overrightarrow{\mathbf{T U}}$ and $\overrightarrow{\mathbf{U P}}$ represent the sides of a regular hexagon.
Statement I $\overrightarrow{\mathbf{P Q}} \times(\overrightarrow{\mathbf{R S}}+\overrightarrow{\mathbf{S T}}) \neq \overrightarrow{\mathbf{0}}$.
because
Statement II $\overrightarrow{\mathbf{P Q}} \times \overrightarrow{\mathbf{R S}}=\overrightarrow{\mathbf{0}}$ and $\overrightarrow{\mathbf{P Q}} \times \overrightarrow{\mathbf{S T}} \neq \overrightarrow{\mathbf{0}}$
(2007, 3M)
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Answer:
Correct Answer: 14. (c)
Solution:
- Since, $\overrightarrow{\mathbf{P Q}}$ is not parallel to $\overrightarrow{\mathbf{T R}}$.
$\because \overrightarrow{\mathbf{T R}}$ is resultant of $\overrightarrow{\mathbf{R S}}$ and $\overrightarrow{\mathbf{S T}}$ vectors. $\Rightarrow \overrightarrow{\mathbf{P Q}} \times(\overrightarrow{\mathbf{R S}}+\overrightarrow{\mathbf{S T}}) \neq \overrightarrow{\mathbf{0}}$.
But for Statement II, we have
$$ \overrightarrow{PQ} \times \overrightarrow{RS}=\overrightarrow{\mathbf{0}} $$
which is not possible as $\overrightarrow{\mathbf{P Q}}$ not parallel to $\overrightarrow{\mathbf{R S}}$.
Hence, Statement I is true and Statement II is false.
