• # question_answer If $\left| \overrightarrow{a}+\overrightarrow{b} \right|=\left| \overrightarrow{a}-\overrightarrow{b} \right|$ then the vectors $\vec{a}$and $\vec{b}$are adjacent sides of A) a rectangle B) a square C) a rhombus D) none of these

 Let $\overrightarrow{OA}=\overrightarrow{a}$ and $\overrightarrow{OB}=\overrightarrow{b}$. Complete the parallelogram OACB. $\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{OA}+\overrightarrow{OB}=\overrightarrow{OC}$ $\Rightarrow \left| \overrightarrow{a}+\overrightarrow{b} \right|=OC$
 Again $\overrightarrow{a}-\overrightarrow{b}=\overrightarrow{OA}-\overrightarrow{OB}=\overrightarrow{BA}$ $\Rightarrow \left| \overrightarrow{a}-\overrightarrow{b} \right|=BA$ Given $\left| \overrightarrow{a}+\overrightarrow{b} \right|=\left| \overrightarrow{a}-\overrightarrow{b} \right|\Rightarrow \operatorname{OC}=BA$ $\therefore$Diagonals of the parallelogram OACB are equal. $\therefore$$\overrightarrow{\operatorname{a}}$and $\overrightarrow{\operatorname{b}}$are adjacent sides of a rectangle.