question_answer

The sum and difference of two vectors are perpendicular to each other. Prove that the vectors are equal in magnitude.

Answer:

                As the vectors \[\vec{A}+\vec{B}\] and \[\vec{A}-\vec{B}\] are perpendicular to each other, therefore \[(\vec{A}+\vec{B}).(\vec{A}-\vec{B})=0\] \[\vec{A}.\vec{A}-\vec{A}.\vec{B}+\vec{B}.\vec{A}-\vec{B}.\vec{B}=0\]                 or  \[{{A}^{2}}-{{B}^{2}}=0\] \[\Rightarrow \] \[A=B\]      \[[\because \vec{A}.\vec{B}=\vec{B}.\vec{A}]\]