What is the angle between `veca` and the resultant of `veca+vecb` and `veca-vecb` ?

Options

A. 0
B. `"tan"^(-1)(a)/(b)`
C. `"tan"^(-1)(b)/(a)`
D. `"tan"^(-1)((a-b))/((a+b))`

Answer

To find the angle between the 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
We need to find the resultant vector \(\vec{R}\) of \(\vec{A} + \vec{B}\) and \(\vec{A} - \vec{B}\).

\[ \vec{R} = (\vec{A} + \vec{B}) + (\vec{A} - \vec{B}) \]

### Step 2: Simplify the Resultant Vector
Now, we simplify the expression for \(\vec{R}\):

\[ \vec{R} = \vec{A} + \vec{B} + \vec{A} - \vec{B} \]

The \(\vec{B}\) terms cancel out:

\[ \vec{R} = 2\vec{A} \]

### Step 3: Determine the Angle Between \(\vec{A}\) and \(\vec{R}\)
Now we need to find the angle \(\theta\) between \(\vec{A}\) and \(\vec{R}\):

\[ \theta = \text{angle between } \vec{A} \text{ and } 2\vec{A} \]

### Step 4: Analyze the Relationship
Since \(\vec{R} = 2\vec{A}\), \(\vec{A}\) and \(\vec{R}\) are parallel vectors. The angle between any vector and a scalar multiple of itself is \(0\) degrees.

### Conclusion
Thus, the angle between \(\vec{A}\) and the resultant \(\vec{R}\) is:

\[ \theta = 0 \text{ degrees} \]

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

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