What is the angle between `vecA` and the resultant of `(vecA + hatB) and (vecA-hat B) `?

Options

A. `0^(@)`
B. `tan^(-1)((A)/(B))`
C. `tan^(-1)((B)/(A))`
D. `tan^(-1)((A-B)/(A+B))`

Explanation

To solve the problem, we need to find the angle between the vector \( \vec{A} \) and the resultant of the vectors \( \vec{A} + \hat{B} \) and \( \vec{A} - \hat{B} \).

### Step-by-Step Solution:

1. **Identify the Vectors**:
We have two vectors:
- \( \vec{A} + \hat{B} \)
- \( \vec{A} - \hat{B} \)

2. **Calculate the Resultant Vector**:
The resultant vector \( \vec{R} \) of the two vectors can be calculated by adding them:
\[ \vec{R} = (\vec{A} + \hat{B}) + (\vec{A} - \hat{B}) \]
Simplifying this, we get:
\[ \vec{R} = \vec{A} + \hat{B} + \vec{A} - \hat{B} = 2\vec{A} \]

3. **Determine the Angle**:
Now we need to find the angle \( \theta \) between \( \vec{A} \) and the resultant vector \( \vec{R} = 2\vec{A} \).
Since \( 2\vec{A} \) is just a scalar multiple of \( \vec{A} \), they are in the same direction.

4. **Conclusion**:
The angle \( \theta \) between \( \vec{A} \) and \( 2\vec{A} \) is \( 0^\circ \) because they are parallel.

### Final Answer:
The angle between \( \vec{A} \) and the resultant of \( (\vec{A} + \hat{B}) \) and \( (\vec{A} - \hat{B}) \) is \( 0^\circ \).

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