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

Options

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

Explanation

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

### Step 1: Define the Resultant Vector
The resultant vector \( \vec{R} \) is defined as:
\[ \vec{R} = (\vec{A} + \vec{B}) + (\vec{A} - \vec{B}) \]

### Step 2: Simplify the Resultant Vector
Now, we simplify \( \vec{R} \):
\[ \vec{R} = \vec{A} + \vec{B} + \vec{A} - \vec{B} = 2\vec{A} \]

### Step 3: Determine the Angle Between \( \vec{A} \) and \( \vec{R} \)
Now, we need to find the angle \( \theta \) between \( \vec{A} \) and \( \vec{R} \):
\[ \vec{R} = 2\vec{A} \]
Since \( \vec{R} \) is just a scaled version of \( \vec{A} \) (specifically, twice the magnitude of \( \vec{A} \)), the angle between \( \vec{A} \) and \( \vec{R} \) is:
\[ \theta = 0^\circ \]

### Conclusion
Thus, the angle between \( \vec{A} \) and the resultant vector \( \vec{R} \) is \( 0^\circ \).

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