A) A = B
B) A = 2B
C) B = 2A
D) \[\mathbf{\vec{A}}\]and \[\mathbf{\vec{B}}\] have the same direction
Correct Answer: A
Solution :
The sum of vectors \[\overset{\to }{\mathop{\mathbf{A}}}\,\] and\[\overset{\to }{\mathop{\mathbf{B}}}\,\] \[\overset{\to }{\mathop{{{\mathbf{R}}_{\mathbf{1}}}}}\,=\overset{\to }{\mathop{\mathbf{A}}}\,+\overset{\to }{\mathop{\mathbf{B}}}\,\] The difference of vectors\[\overset{\to }{\mathop{\mathbf{A}}}\,\] and\[\overset{\to }{\mathop{\mathbf{B}}}\,\] \[\overset{\to }{\mathop{{{\mathbf{R}}_{\mathbf{2}}}}}\,=\overset{\to }{\mathop{\mathbf{A}}}\,-\overset{\to }{\mathop{\mathbf{B}}}\,\] Since, \[\overset{\to }{\mathop{{{\mathbf{R}}_{\mathbf{1}}}}}\,\] and \[\overset{\to }{\mathop{{{\mathbf{R}}_{\mathbf{2}}}}}\,\] are at right angle, their dot product will be zero,\[ie\], \[\overset{\to }{\mathop{{{\mathbf{R}}_{\mathbf{1}}}}}\,\cdot \overset{\to }{\mathop{{{\mathbf{R}}_{\mathbf{2}}}}}\,=(\overset{\to }{\mathop{\mathbf{A}}}\,+\overset{\to }{\mathop{\mathbf{B}}}\,)\cdot (\overset{\to }{\mathop{\mathbf{A}}}\,-\overset{\to }{\mathop{\mathbf{B}}}\,)\] or =\[0=\overset{\to }{\mathop{\mathbf{A}}}\,\cdot \overset{\to }{\mathop{\mathbf{A}}}\,-\overset{\to }{\mathop{\mathbf{A}}}\,\cdot \overset{\to }{\mathop{\mathbf{B}}}\,+\overset{\to }{\mathop{\mathbf{B}}}\,\cdot \overset{\to }{\mathop{\mathbf{A}}}\,-\overset{\to }{\mathop{\mathbf{B}}}\,\cdot \overset{\to }{\mathop{\mathbf{B}}}\,\] or \[0={{A}^{2}}-{{B}^{2}}\] (as\[\overset{\to }{\mathop{\mathbf{A}}}\,\cdot \overset{\to }{\mathop{\mathbf{B}}}\,=\overset{\to }{\mathop{\mathbf{B}}}\,\cdot \overset{\to }{\mathop{\mathbf{A}}}\,\]) \[\therefore \] \[{{A}^{2}}={{B}^{2}}\] or \[A=B\]
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