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 \( P + Q \) and \( P - Q \), we can use the concept of the dot product of vectors. Here’s a step-by-step solution:

### Step 1: Define the angle
Let \( \alpha \) be the angle between the vectors \( P + Q \) and \( P - Q \).

### Step 2: Use the dot product formula
The dot product of two vectors \( A \) and \( B \) is given by:
\[ A \cdot B = |A| |B| \cos \theta \]
In our case, we have:
\[ (P + Q) \cdot (P - Q) = |P + Q| |P - Q| \cos \alpha \]

### Step 3: Calculate the dot product
Now, we can calculate \( (P + Q) \cdot (P - Q) \):
\[ (P + Q) \cdot (P - Q) = P \cdot P - P \cdot Q + Q \cdot P - Q \cdot Q \]
Since \( P \cdot Q = Q \cdot P \), this simplifies to:
\[ = |P|^2 - |Q|^2 \]

### Step 4: Calculate the magnitudes
Next, we need to find the magnitudes \( |P + Q| \) and \( |P - Q| \):
1. For \( |P + Q| \):
\[ |P + Q| = \sqrt{|P|^2 + |Q|^2 + 2P \cdot Q} \]
2. For \( |P - Q| \):
\[ |P - Q| = \sqrt{|P|^2 + |Q|^2 - 2P \cdot Q} \]

### Step 5: Substitute back into the equation
Now substituting back into the equation for the dot product:
\[ |P|^2 - |Q|^2 = |P + Q| |P - Q| \cos \alpha \]

### Step 6: Analyze the cases
Now we analyze the cases based on the magnitudes of \( P \) and \( Q \):
1. **Case 1**: If \( |P| > |Q| \), then \( |P|^2 - |Q|^2 > 0 \) implies \( \cos \alpha > 0 \), hence \( \alpha \) is acute (between \( 0^\circ \) and \( 90^\circ \)).
2. **Case 2**: If \( |P| < |Q| \), then \( |P|^2 - |Q|^2 < 0 \) implies \( \cos \alpha < 0 \), hence \( \alpha \) is obtuse (between \( 90^\circ \) and \( 180^\circ \)).
3. **Case 3**: If \( |P| = |Q| \), then \( |P|^2 - |Q|^2 = 0 \) which leads to an indeterminate form, meaning the angle cannot be defined.

### Conclusion
From the analysis, we conclude that the angle \( \alpha \) between \( P + Q \) and \( P - Q \) can vary between \( 0^\circ \) and \( 180^\circ \) depending on the magnitudes of \( P \) and \( Q \).

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

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