What is the angle between P and Q the cross product of `(P+Q)and (P-Q)`?

Options

A. `90^(@)`
B. `tan^(-1) (p//Q)`
C. `tan ^(-1)(Q//p)`
D. `0^(@)`

Explanation

To find the angle between the vectors \( P \) and \( Q \) in the context of the cross product of \( (P + Q) \) and \( (P - Q) \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Cross Product**:
The cross product of two vectors \( A \) and \( B \) is given by:
\[ A \times B = |A||B| \sin(\theta) \hat{n} \]
where \( \theta \) is the angle between the vectors \( A \) and \( B \), and \( \hat{n} \) is the unit vector perpendicular to the plane formed by \( A \) and \( B \).

2. **Identify the Vectors**:
We need to compute the cross product \( (P + Q) \times (P - Q) \).

3. **Use the Distributive Property of Cross Product**:
The cross product can be expanded as follows:
\[ (P + Q) \times (P - Q) = P \times P + P \times (-Q) + Q \times P + Q \times (-Q) \]
Since the cross product of any vector with itself is zero, we have:
\[ P \times P = 0 \quad \text{and} \quad Q \times Q = 0 \]
Thus, the expression simplifies to:
\[ (P + Q) \times (P - Q) = 0 - P \times Q + Q \times P \]

4. **Recognize the Anti-Symmetry of the Cross Product**:
The cross product is anti-symmetric, meaning:
\[ Q \times P = - (P \times Q) \]
Therefore, we can rewrite the expression as:
\[ (P + Q) \times (P - Q) = - P \times Q - P \times Q = -2(P \times Q) \]

5. **Determine the Angle**:
The result \( -2(P \times Q) \) indicates that the direction of the resultant vector is perpendicular to both \( P \) and \( Q \). Since the cross product results in a vector that is perpendicular to the plane formed by \( P \) and \( Q \), the angle between \( P \) and \( Q \) must be \( 90^\circ \).

### Final Answer:
The angle between the vectors \( P \) and \( Q \) is \( 90^\circ \).