A) are not equal to each other in magnitude
B) cannot be predicted
C) are equal to each other
D) are equal to each other in magnitude
Correct Answer: D
Solution :
Key Idea: The two vectors must be perpendicular if their dot product must be zero. Let\[\overrightarrow{A}\]and\[\overrightarrow{B}\]be two forces. The sum of the two forces, \[{{\overrightarrow{F}}_{1}}=\overrightarrow{A}+\overrightarrow{B}\] ...(i) The difference of the two forces, \[{{\overrightarrow{F}}_{2}}=\overrightarrow{A}-\overrightarrow{B}\] ...(ii) Since, sum of the two forces is perpendicular to their difference as said, so \[{{\overrightarrow{F}}_{1}}.{{\overrightarrow{F}}_{2}}=0\] Or \[(\overrightarrow{A}+\overrightarrow{B}).(\overrightarrow{A}-\overrightarrow{B})=0\] Or \[{{A}^{2}}-\overrightarrow{A}.\overrightarrow{B}+\overrightarrow{B}.\overrightarrow{A}-{{B}^{2}}=0\] Or \[{{A}^{2}}={{B}^{2}}\] Or \[|\overrightarrow{A}|=|\overrightarrow{B}|\] Thus, the forces are equal to each other in magnitude.
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