Angle Between Equal Magnitude Vectors with Equal Sum and Resultant

Understanding the relationship between vectors with equal magnitude and their resultant is a fundamental concept in mathematics and physics. In this article, we will explore the specific scenario where the sum of the magnitudes of two vectors with equal magnitude is equal to the magnitude of their resultant vector.

Problem Statement

The problem at hand involves two vectors, A and B, both with equal magnitudes (#916;A #916;B m). The challenge is to determine the angle, , between them such that the magnitude of their resultant vector (R) equals the sum of their magnitudes.

Solution Approach

To solve this problem, we will follow a step-by-step approach:

Step 1: Finding the Magnitude of the Resultant Vector

The magnitude of the resultant vector (R) can be found using the vector addition formula:

[ R sqrt{A^2 B^2 2AB cos theta} ]

Since both A and B have equal magnitudes (m), we substitute A and B with m:

[ R sqrt{m^2 m^2 2m^2 cos theta} sqrt{2m^2 (1 cos theta)} m sqrt{2(1 cos theta)} ]

Step 2: Setting the Condition

The sum of the magnitudes of vectors A and B is:

[ A B m m 2m ]

We set the magnitude of the resultant vector (R) equal to the sum of the magnitudes:

[ m sqrt{2(1 cos theta)} 2m ]

Dividing both sides by m (assuming m is not zero):

[ sqrt{2(1 cos theta)} 2 ]

Step 3: Solving for

Squaring both sides of the equation:

[ 2(1 cos theta) 4 ]

Dividing by 2:

[ 1 cos theta 2 ]

Subtracting 1:

[ cos theta 1 ]

The angle for which is:

[ theta 0^circ ]

Conclusion

The angle between the two vectors is , indicating that the vectors are pointing in the same direction. The resultant vector's magnitude equals the sum of the magnitudes when the two vectors are aligned.

Additional Insight: Parallelogram Method

For those familiar with the parallelogram method of vector addition and subtraction, the solution becomes even clearer. When the two vectors are of the same length, there is an angle at which the resultant vector (red arrow) and the difference vector (blue arrow) have the same length. Drawing this on paper will illustrate the concept easily.

By understanding these principles, you can apply them to more complex problems in vector analysis and related fields.