**The Angle between a Vector and its Resultant**

When two vectors of equal magnitude, represented by vector A and vector B, are added together, the result is a vector called the resultant vector or vector R. The angle between vector A and vector B is denoted as theta (θ).

To find the angle between vector A or vector B and their resultant vector R, we need to consider the geometry of vector addition and the properties of triangles.

**Step 1: Determine the Magnitude of the Resultant Vector**

Before finding the angle, we need to calculate the magnitude of the resultant vector R. The magnitude of R can be obtained using the vector addition formula:

|R| = √(A^2 + B^2 + 2ABcosθ)

**Step 2: Apply the Law of Cosines**

Next, we can use the law of cosines to find the angle between vector A or vector B and the resultant vector R. The law of cosines states:

c^2 = a^2 + b^2 - 2abcosC

In this case, vector A or vector B acts as the side "a" of the triangle, the resultant vector R acts as the side "c," and the angle between them (θ) acts as angle "C." Therefore, we can rewrite the equation as:

|R|^2 = |A|^2 + |B|^2 - 2|A||B|cosθ

Substituting the magnitude values, we get:

(A^2 + B^2 + 2ABcosθ) = A^2 + B^2 - 2ABcosθ

Simplifying the equation, we find:

4ABcosθ = 0

Since the magnitude of vectors A and B is non-zero, we can conclude that cosθ = 0.

**Step 3: Solve for the Angle**

To find the angle θ, we need to determine the values of cosθ that satisfy the equation cosθ = 0. The values of cosθ that satisfy this equation are θ = 90° or θ = 270°.

Therefore, the angle between vector A or vector B and their resultant vector R is either 90° or 270°.

**Conclusion**

The angle between a vector A or vector B and their resultant vector R, when the vectors have equal magnitude and θ is the angle between them, is either 90° or 270°. This conclusion is derived from the properties of vector addition, the law of cosines, and the condition cosθ = 0.