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The vector sum of two forces is perpendicular to their vector differences. In that case, the forces[2003]
- a)can not be pr edicted
- b)are equal to each other
- c)are equal to each other in magnitude
- d)are not equal to each other in magnitude
Correct answer is option 'C'. Can you explain this answer?
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Explanation:
To understand why the vector sum of two forces is equal to each other in magnitude, we need to first understand the concept of vector addition and subtraction.
Vector addition: When two forces act on an object in the same direction, their magnitudes are simply added together. For example, if two forces of 5N and 3N act in the same direction, the resultant force would be 8N.
Vector subtraction: When two forces act on an object in opposite directions, their magnitudes are subtracted. For example, if a force of 5N acts to the right and a force of 3N acts to the left, the resultant force would be 2N to the right.
The vector sum of two forces is perpendicular to their vector differences:
Now, let's consider two forces, F1 and F2, and their vector sum, F_sum, and vector difference, F_diff.
F_sum = F1 + F2
F_diff = F1 - F2
If the vector sum, F_sum, is perpendicular to the vector difference, F_diff, it means that their dot product is zero.
F_sum · F_diff = 0
Using the definitions of vector addition and subtraction:
(F1 + F2) · (F1 - F2) = 0
Expanding the dot product:
(F1 · F1) + (F1 · -F2) + (F2 · F1) + (F2 · -F2) = 0
Since the dot product of a vector with itself is the square of its magnitude:
|F1|^2 - (F1 · F2) + (F2 · F1) - |F2|^2 = 0
Simplifying:
|F1|^2 - |F2|^2 = 0
This equation shows that the magnitudes of the two forces, F1 and F2, are equal to each other. Therefore, the correct answer is option C - the forces are equal to each other in magnitude.
To understand why the vector sum of two forces is equal to each other in magnitude, we need to first understand the concept of vector addition and subtraction.
Vector addition: When two forces act on an object in the same direction, their magnitudes are simply added together. For example, if two forces of 5N and 3N act in the same direction, the resultant force would be 8N.
Vector subtraction: When two forces act on an object in opposite directions, their magnitudes are subtracted. For example, if a force of 5N acts to the right and a force of 3N acts to the left, the resultant force would be 2N to the right.
The vector sum of two forces is perpendicular to their vector differences:
Now, let's consider two forces, F1 and F2, and their vector sum, F_sum, and vector difference, F_diff.
F_sum = F1 + F2
F_diff = F1 - F2
If the vector sum, F_sum, is perpendicular to the vector difference, F_diff, it means that their dot product is zero.
F_sum · F_diff = 0
Using the definitions of vector addition and subtraction:
(F1 + F2) · (F1 - F2) = 0
Expanding the dot product:
(F1 · F1) + (F1 · -F2) + (F2 · F1) + (F2 · -F2) = 0
Since the dot product of a vector with itself is the square of its magnitude:
|F1|^2 - (F1 · F2) + (F2 · F1) - |F2|^2 = 0
Simplifying:
|F1|^2 - |F2|^2 = 0
This equation shows that the magnitudes of the two forces, F1 and F2, are equal to each other. Therefore, the correct answer is option C - the forces are equal to each other in magnitude.
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The vector sum of two forces is perpendicular to their vector differences. In that case, the forces[2003]a)can not be pr edictedb)are equal to each otherc)are equal to each other in magnituded)are not equal to each other in magnitudeCorrect answer is option 'C'. Can you explain this answer?
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