What is the angle between `vec(P)` and the resultant of `(vec(P)+vec(Q))` and `(vec(P)-vec(Q))` ?

Options

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

Explanation

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

### Step-by-Step Solution:

1. **Define the Resultant Vector**:
The resultant vector \( \vec{R} \) of the vectors \((\vec{P} + \vec{Q})\) and \((\vec{P} - \vec{Q})\) can be expressed as:
\[ \vec{R} = (\vec{P} + \vec{Q}) + (\vec{P} - \vec{Q}) \]

2. **Combine the Vectors**:
Simplifying the expression for \( \vec{R} \):
\[ \vec{R} = \vec{P} + \vec{Q} + \vec{P} - \vec{Q} = 2\vec{P} \]

3. **Identify the Vectors**:
Now we have:
- \(\vec{P}\)
- \(\vec{R} = 2\vec{P}\)

4. **Determine the Angle Between the Vectors**:
The angle \( \theta \) between \(\vec{P}\) and \(2\vec{P}\) can be determined. Since \(2\vec{P}\) is simply \(\vec{P}\) scaled by a factor of 2, both vectors point in the same direction.

5. **Calculate the Angle**:
The angle between two vectors that point in the same direction is:
\[ \theta = 0^\circ \]

### Final Answer:
The angle between \(\vec{P}\) and the resultant vector \( \vec{R} \) is \(0^\circ\).

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