What can be the angle between `(vec(P) + vec(Q))` and `(vec(P) - vec(Q))` ?

Options

A. `90^(@)`
B. `0^(@)` only
C. any angle between `0^(@)` and `180^(@)`
D. `180^(@)`

Answer

To find the angle between the vectors \(\vec{P} + \vec{Q}\) and \(\vec{P} - \vec{Q}\), we can use the concept of the dot product of vectors.

### Step-by-Step Solution:

1. **Understanding the Dot Product**:
The dot product of two vectors \(\vec{A}\) and \(\vec{B}\) is given by:
\[ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta) \]
where \(\theta\) is the angle between the two vectors.

2. **Setting Up the Problem**:
Let \(\vec{A} = \vec{P} + \vec{Q}\) and \(\vec{B} = \vec{P} - \vec{Q}\). We want to find the angle \(\alpha\) between \(\vec{A}\) and \(\vec{B}\).

3. **Calculating the Dot Product**:
We compute the dot product \(\vec{A} \cdot \vec{B}\):
\[ \vec{A} \cdot \vec{B} = (\vec{P} + \vec{Q}) \cdot (\vec{P} - \vec{Q}) \]
Expanding this, we get:
\[ = \vec{P} \cdot \vec{P} - \vec{P} \cdot \vec{Q} + \vec{Q} \cdot \vec{P} - \vec{Q} \cdot \vec{Q} \]
Since \(\vec{P} \cdot \vec{Q} = \vec{Q} \cdot \vec{P}\), this simplifies to:
\[ = |\vec{P}|^2 - |\vec{Q}|^2 \]

4. **Finding Magnitudes**:
Next, we need to find the magnitudes of \(\vec{A}\) and \(\vec{B}\):
- The magnitude of \(\vec{A}\):
\[ |\vec{A}| = |\vec{P} + \vec{Q}| \]
- The magnitude of \(\vec{B}\):
\[ |\vec{B}| = |\vec{P} - \vec{Q}| \]

5. **Using the Dot Product Formula**:
Now, we can express the cosine of the angle \(\alpha\) as:
\[ \cos(\alpha) = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|} = \frac{|\vec{P}|^2 - |\vec{Q}|^2}{|\vec{A}| |\vec{B}|} \]

6. **Analyzing the Cases**:
- **Case 1**: If \(|\vec{P}| > |\vec{Q}|\), then \(\cos(\alpha) > 0\) which implies \(0 < \alpha < 90^\circ\).
- **Case 2**: If \(|\vec{P}| < |\vec{Q}|\), then \(\cos(\alpha) < 0\) which implies \(90^\circ < \alpha < 180^\circ\).
- **Case 3**: If \(|\vec{P}| = |\vec{Q}|\), then \(\cos(\alpha) = 0\) which implies \(\alpha = 90^\circ\).

7. **Conclusion**:
Therefore, the angle \(\alpha\) between the vectors \(\vec{P} + \vec{Q}\) and \(\vec{P} - \vec{Q}\) can be any angle between \(0^\circ\) and \(180^\circ\).

### Final Answer:
The angle between \((\vec{P} + \vec{Q})\) and \((\vec{P} - \vec{Q})\) can be any angle between \(0^\circ\) and \(180^\circ\).