Let the vectors overline{PQ}, overline{QR}, o | vedclass

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Question

(C) In a closed hexagon $PQRSTU$,the sum of the vectors representing its sides is zero:$\overline{PQ} + \overline{QR} + \overline{RS} + \overline{ST}

Vedclass · Accepted Answer

Statement-$1$ is true,Statement-$2$ is false.

Vedclass · Answer

(C) In a closed hexagon $PQRSTU$,the sum of the vectors representing its sides is zero:$\overline{PQ} + \overline{QR} + \overline{RS} + \overline{ST} + \overline{TU} + \overline{UP} = \vec{0}$.Statement-$2$ claims $\overline{PQ} 	imes \overline{RS} = \vec{0}$ and $\overline{PQ} 	imes \overline{ST} = \vec{0}$. This implies that $\overline{PQ}$ is parallel to $\overline{RS}$ and $\overline{PQ}$ is parallel to $\overline{ST}$. If $\overline{PQ} \parallel \overline{RS}$ and $\overline{PQ} \parallel \overline{ST}$,then $\overline{RS}$ and $\overline{ST}$ must be parallel to each other. This is not generally true for any arbitrary hexagon. Thus,Statement-$2$ is false.For Statement-$1$,$\overline{PQ} 	imes (\overline{RS} + \overline{ST}) = \overline{PQ} 	imes \overline{RS} + \overline{PQ} 	imes \overline{ST}$. Since the sides of a general hexagon are not necessarily parallel,this cross product is generally non-zero. Thus,Statement-$1$ is true.

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