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  4. Let the vectors P Q, Q R, R S, S T, T U and U P, represent the sides of a regular hexagon. STATEMENT-1: P Q ×(R S+S T) ≠ 0. because STATEMENT-2:P Q × R S=0 and P Q × S T ≠ 0

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Q. Let the vectors $\overrightarrow{P Q}, \overrightarrow{Q R}, \overrightarrow{R S}, \overrightarrow{S T}, \overrightarrow{T U}$ and $\overrightarrow{U P}$, represent the sides of a regular hexagon.
STATEMENT-1: $\overrightarrow{P Q} \times(\overrightarrow{R S}+\overrightarrow{S T}) \neq \overrightarrow{0}$.
because
STATEMENT-2:$\overrightarrow{P Q} \times \overrightarrow{R S}=\overrightarrow{0}$ and $\overrightarrow{P Q} \times \overrightarrow{S T} \neq \overrightarrow{0}$

JEE AdvancedJEE Advanced 2007

Solution:

Statement-1: $\overrightarrow{P Q} \times(\overrightarrow{R S} \times \overrightarrow{S T}) \neq 0 .$
Since $\overrightarrow{P Q}$ is not parallel to $\overrightarrow{R T}$, we get
$\overrightarrow{P Q} \times(\overrightarrow{R T})$
That is, Statement-2 is true.
image
$\overrightarrow{P Q} \times(\overrightarrow{R T}) \neq 0$
Statement-2: $\overrightarrow{P Q} \times \overrightarrow{R S}=\overrightarrow{0}$ and $\overrightarrow{P Q} \times \overrightarrow{S T} \neq \overrightarrow{0}$
Since $\overrightarrow{P Q}$ is not parallel to $\overrightarrow{R S}$, Statement-2 is false.