The angles between P+Q and P-Q will be

Options

A. `90^@`
B. between `0^@ and 180^@`
C. `180^@` only
D. None of these

Explanation

To find the angle between the vectors \( \mathbf{P} + \mathbf{Q} \) and \( \mathbf{P} - \mathbf{Q} \), we can use the dot product of the two vectors. Here’s the step-by-step solution:

### Step 1: Write the expression for the dot product
We start by calculating the dot product of the two vectors:
\[ (\mathbf{P} + \mathbf{Q}) \cdot (\mathbf{P} - \mathbf{Q}) \]

### Step 2: Expand the dot product
Using the distributive property of the dot product, we can expand this expression:
\[ \mathbf{P} \cdot \mathbf{P} - \mathbf{P} \cdot \mathbf{Q} + \mathbf{Q} \cdot \mathbf{P} - \mathbf{Q} \cdot \mathbf{Q} \]
This simplifies to:
\[ \mathbf{P}^2 - \mathbf{Q}^2 \]
since \( \mathbf{P} \cdot \mathbf{Q} = \mathbf{Q} \cdot \mathbf{P} \).

### Step 3: Analyze the result
The result \( \mathbf{P}^2 - \mathbf{Q}^2 \) can be positive, negative, or zero depending on the magnitudes of \( \mathbf{P} \) and \( \mathbf{Q} \):
- If \( \mathbf{P} > \mathbf{Q} \), then \( \mathbf{P}^2 - \mathbf{Q}^2 > 0 \).
- If \( \mathbf{P} < \mathbf{Q} \), then \( \mathbf{P}^2 - \mathbf{Q}^2 < 0 \).
- If \( \mathbf{P} = \mathbf{Q} \), then \( \mathbf{P}^2 - \mathbf{Q}^2 = 0 \).

### Step 4: Determine the angle
The cosine of the angle \( \theta \) between the two vectors is given by:
\[ \cos \theta = \frac{(\mathbf{P} + \mathbf{Q}) \cdot (\mathbf{P} - \mathbf{Q})}{|\mathbf{P} + \mathbf{Q}| |\mathbf{P} - \mathbf{Q}|} \]
Since \( \mathbf{P}^2 - \mathbf{Q}^2 \) can take any value (positive, negative, or zero), the angle \( \theta \) can be:
- \( 90^\circ \) if \( \mathbf{P}^2 - \mathbf{Q}^2 = 0 \)
- Between \( 0^\circ \) and \( 180^\circ \) if \( \mathbf{P}^2 - \mathbf{Q}^2 \) is positive or negative.

### Conclusion
Thus, the angle between \( \mathbf{P} + \mathbf{Q} \) and \( \mathbf{P} - \mathbf{Q} \) can be any angle between \( 0^\circ \) and \( 180^\circ \).

### Final Answer
The angles between \( \mathbf{P} + \mathbf{Q} \) and \( \mathbf{P} - \mathbf{Q} \) will be between \( 0^\circ \) and \( 180^\circ \).

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