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 \[\vec{B}\] \[{{\vec{R}}_{2}}={{\vec{A}}_{1}}-\vec{B}\] 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 \[0=\vec{A}\,\,.\,\vec{A}-\vec{A}\,.\,\,\vec{B}+\vec{B}\,\,.\,\,\vec{A}-\vec{B}\,\,.\,\,\vec{B}\] 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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