What is the angle between A and the resultant of `(vec A + vec B) and (vec A -vec B)` ?

Options

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

Explanation

To find the angle between vector A and the resultant of (A + B) and (A - B), we can follow these steps:

### Step 1: Define the Vectors
Let vector \( \vec{A} \) and vector \( \vec{B} \) be given. We need to find the resultant of \( \vec{A} + \vec{B} \) and \( \vec{A} - \vec{B} \).

### Step 2: Calculate the Resultant Vectors
1. **Resultant of \( \vec{A} + \vec{B} \)**:
\[ \vec{R_1} = \vec{A} + \vec{B} \]

2. **Resultant of \( \vec{A} - \vec{B} \)**:
\[ \vec{R_2} = \vec{A} - \vec{B} \]

### Step 3: Find the Resultant of \( \vec{R_1} \) and \( \vec{R_2} \)
To find the resultant of \( \vec{R_1} \) and \( \vec{R_2} \):
\[ \vec{R} = \vec{R_1} + \vec{R_2} = (\vec{A} + \vec{B}) + (\vec{A} - \vec{B}) \]
This simplifies to:
\[ \vec{R} = \vec{A} + \vec{B} + \vec{A} - \vec{B} = 2\vec{A} \]

### Step 4: Determine the Angle Between \( \vec{A} \) and \( \vec{R} \)
The resultant \( \vec{R} \) is \( 2\vec{A} \). Since \( \vec{R} \) is a scalar multiple of \( \vec{A} \), they are parallel.

### Conclusion
The angle between \( \vec{A} \) and \( \vec{R} \) is:
\[ \text{Angle} = 0^\circ \]

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

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