A) \[A=B\]
B) \[A=2B\]
C) \[B=2A\]
D) \[\vec{A}\] and \[\vec{B}\] have the same direction
Correct Answer: A
Solution :
Key Idea If two vectors are at right angles, then their dot product will be zero. The sum of vectors\[\vec{A}\] and \[\vec{B}\] \[\vec{R}{{ }_{1}}=\vec{A}+\vec{B}\] The difference of vectors \[\vec{A}\]and \[\vec{B}\] \[\vec{R}{{ }_{2}}=\vec{A}-\vec{B}\] Since, \[{{\vec{R}}_{1}}\]and \[{{\vec{R}}_{2}}\]are at right angles, their dot product will be zero, ie, \[{{\vec{R}}_{1}}.{{\vec{R}}_{2}}=(\vec{A}+\vec{B}).(\vec{A}-\vec{B})\] or \[0=\vec{A}.\vec{A}-\vec{A}.\vec{B}+\vec{B}+\vec{B}.\vec{A}-\vec{B}.\vec{B}\] or \[0={{A}^{2}}-{{B}^{2}}\] (as \[\vec{A}.\vec{B}=\vec{B}.\vec{A}\]) \[\therefore \] \[{{A}^{2}}={{B}^{2}}\]or \[A=B\]
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