The resultant R of vector P andQ is perpendicular to P and R=P both , then angle betwwen `|P|and|Q| `is

Options

A. `45^(@)`
B. `135^(@)`
C. `120^(@)`
D. All of these

Explanation

To solve the problem step by step, we need to analyze the given conditions about the vectors \( \mathbf{P} \) and \( \mathbf{Q} \).

### Step 1: Understand the Given Information
We are given that the resultant vector \( \mathbf{R} \) of vectors \( \mathbf{P} \) and \( \mathbf{Q} \) is perpendicular to \( \mathbf{P} \) and that \( \mathbf{R} = \mathbf{P} \).

### Step 2: Set Up the Vectors
Let:
- \( \mathbf{P} \) be represented as \( P \) (magnitude of vector \( \mathbf{P} \)).
- \( \mathbf{Q} \) be represented as \( Q \) (magnitude of vector \( \mathbf{Q} \)).
- The angle between \( \mathbf{P} \) and \( \mathbf{Q} \) be \( \theta \).

### Step 3: Use the Condition of Perpendicularity
Since \( \mathbf{R} \) is perpendicular to \( \mathbf{P} \), we can express \( \mathbf{R} \) in terms of its components:
- The x-component of \( \mathbf{R} \) is given by the x-component of \( \mathbf{P} \) plus the x-component of \( \mathbf{Q} \).
- The y-component of \( \mathbf{R} \) is given by the y-component of \( \mathbf{Q} \) (since the y-component of \( \mathbf{P} \) is zero if we assume it is along the x-axis).

### Step 4: Write the Components
Assuming \( \mathbf{P} \) is along the x-axis:
- \( \mathbf{P} = P \hat{i} \)
- \( \mathbf{Q} = Q \cos(\theta) \hat{i} + Q \sin(\theta) \hat{j} \)

The resultant vector \( \mathbf{R} \) can be expressed as:
\[ \mathbf{R} = \mathbf{P} + \mathbf{Q} = P \hat{i} + (Q \cos(\theta) \hat{i} + Q \sin(\theta) \hat{j}) \]
\[ \mathbf{R} = (P + Q \cos(\theta)) \hat{i} + (Q \sin(\theta)) \hat{j} \]

### Step 5: Use the Condition \( \mathbf{R} = \mathbf{P} \)
Since \( \mathbf{R} = \mathbf{P} \), we have:
\[ (P + Q \cos(\theta)) = P \quad \text{(for the x-component)} \]
\[ Q \sin(\theta) = 0 \quad \text{(for the y-component)} \]

### Step 6: Solve the Equations
From \( Q \sin(\theta) = 0 \), we can conclude that either \( Q = 0 \) or \( \sin(\theta) = 0 \). However, since \( \mathbf{Q} \) is a vector, we assume \( Q \neq 0 \), thus \( \sin(\theta) = 0 \) implies \( \theta = 0^\circ \) or \( \theta = 180^\circ \).

### Step 7: Find the Angle Between \( \mathbf{P} \) and \( \mathbf{Q} \)
Now, from \( P + Q \cos(\theta) = P \):
\[ Q \cos(\theta) = 0 \]
Since \( Q \neq 0 \), we have \( \cos(\theta) = 0 \), which means \( \theta = 90^\circ \) or \( \theta = 270^\circ \).

### Step 8: Determine the Final Angle
The angle between \( \mathbf{P} \) and \( \mathbf{Q} \) can be calculated as:
\[ \text{Angle between } \mathbf{P} \text{ and } \mathbf{Q} = 90^\circ + 45^\circ = 135^\circ \]

### Conclusion
Therefore, the angle between \( |\mathbf{P}| \) and \( |\mathbf{Q}| \) is \( 135^\circ \).