When we consider the scenario where the resultant of two vectors A and B is rotated through 90 degrees, and then one of these vectors is reversed, we can analyze the implications of this situation. The question asks us to determine the relationship between the magnitudes of these vectors. Let's break this down step by step.
Understanding Vector Addition
Vectors are quantities that have both magnitude and direction. When we add two vectors, the resultant vector is determined by both the magnitudes of the vectors and the angle between them. If we denote the vectors as A and B, the resultant vector R can be expressed as:
R = A + B
Effect of Rotating the Resultant
When we rotate the resultant vector R by 90 degrees, we are essentially changing its direction while keeping its magnitude the same. This rotation can have significant implications for the original vectors A and B, especially when one of them is reversed.
Reversing One Vector
Let's say we reverse vector A, making it -A. The new resultant vector R' can be expressed as:
R' = -A + B
Analyzing the Resultant Vectors
For the resultant vector R to be rotated into R', we can visualize this geometrically. The original vectors A and B must be positioned such that when A is reversed, the new resultant R' still maintains a relationship with the original resultant R. This leads us to consider the magnitudes of A and B.
Magnitude Relationships
For the rotation of the resultant vector to hold true, the magnitudes of A and B must be in a specific ratio. If we denote the magnitudes of A and B as |A| and |B| respectively, we can derive the necessary condition for the vectors:
- If |A| = |B|, then the rotation will not yield the required 90-degree relationship.
- If |A| is greater than |B|, the resultant will not align correctly after reversing A.
- However, if |A| and |B| are in the ratio of √2:1, the resultant can be manipulated to achieve the desired rotation.
Conclusion on Magnitudes
Thus, the correct relationship between the magnitudes of vectors A and B, given the conditions of the problem, is that they must be in the ratio of √2:1. This ensures that when one vector is reversed, the resultant can still be rotated by 90 degrees, maintaining the geometric integrity of the vectors involved.
In summary, the answer to your question is that the magnitudes of vectors A and B must be in the ratio of √2:1 for the conditions described to hold true. This relationship allows for the necessary geometric transformations while preserving the properties of vector addition.
