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 \[\vec{A}\] and \[\vec{B}\] are two forces. The sum of the two forces. |
| \[{{\vec{F}}_{1}}=\vec{A}+\vec{B}\] (i) |
| The difference of the two forces, |
| \[{{\vec{F}}_{2}}=\vec{A}-\vec{B}\] (ii) |
| Since, sum of the two forces is perpendicular to their differences as given, so |
| \[{{\vec{F}}_{1}}\,.\,{{\vec{F}}_{2}}=0\] |
| or \[(\vec{A}+\vec{B})\,.\,(\vec{A}-\vec{B})=0\] |
| or \[{{A}^{2}}-\vec{A}\,.\,\vec{B}\,+\vec{B}\,.\,\vec{A}-{{B}^{2}}=0\] |
| or \[{{A}^{2}}={{B}^{2}}\] |
| or \[|\vec{A}|\,=\,|\vec{B}|\] |
| Thus, the forces are equal to each other in magnitude. |
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