If O be the origin and if `P(x_1, y_1) and P_2 (x_2, y_2)` are two points, the `OP_1 (OP_2) sin angle P_1OP_2`, is equal to

Options

A. `x_(1)y_(2)+x_(2)y_(1)`
B. `(x_(1)^(2)+y_(1)^(2))(x_(2)^(2)+y_(2)^(2))`
C. `(x_(1)+x_(2))^(2)+(y_(1)+y_(2))^(2)`
D. `x_(1)x_(2)+y_(1)y_(2)`

Answer

To solve the problem, we need to find the expression for \( OP_1 \cdot OP_2 \cdot \sin(\angle P_1OP_2) \) where \( O \) is the origin, \( P_1(x_1, y_1) \), and \( P_2(x_2, y_2) \).

### Step-by-Step Solution:

1. **Identify the Points and Distances**:
- The origin \( O \) is at \( (0, 0) \).
- The point \( P_1 \) has coordinates \( (x_1, y_1) \).
- The point \( P_2 \) has coordinates \( (x_2, y_2) \).
- The distance \( OP_1 \) is given by:
\[ OP_1 = \sqrt{x_1^2 + y_1^2} \]
- The distance \( OP_2 \) is given by:
\[ OP_2 = \sqrt{x_2^2 + y_2^2} \]

2. **Find the Slopes**:
- The slope of line segment \( OP_1 \) is:
\[ m_1 = \frac{y_1 - 0}{x_1 - 0} = \frac{y_1}{x_1} \]
- The slope of line segment \( OP_2 \) is:
\[ m_2 = \frac{y_2 - 0}{x_2 - 0} = \frac{y_2}{x_2} \]

3. **Calculate the Tangent of the Angle**:
- The tangent of the angle \( \angle P_1OP_2 \) can be calculated using the formula:
\[ \tan(\angle P_1OP_2) = \frac{m_2 - m_1}{1 + m_1 m_2} \]
- Substituting the values of \( m_1 \) and \( m_2 \):
\[ \tan(\angle P_1OP_2) = \frac{\frac{y_2}{x_2} - \frac{y_1}{x_1}}{1 + \frac{y_1 y_2}{x_1 x_2}} = \frac{y_2 x_1 - y_1 x_2}{x_1 x_2 + y_1 y_2} \]

4. **Calculate the Sine of the Angle**:
- From the relationship between sine and tangent, we know:
\[ \sin(\angle P_1OP_2) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{x_1 y_2 - y_1 x_2}{\sqrt{(x_1 y_2 - y_1 x_2)^2 + (x_1 x_2 + y_1 y_2)^2}} \]

5. **Combine Everything**:
- Now we can combine the distances and the sine value:
\[ OP_1 \cdot OP_2 \cdot \sin(\angle P_1OP_2) = \sqrt{x_1^2 + y_1^2} \cdot \sqrt{x_2^2 + y_2^2} \cdot \frac{x_1 y_2 - y_1 x_2}{\sqrt{(x_1 y_2 - y_1 x_2)^2 + (x_1 x_2 + y_1 y_2)^2}} \]
- This simplifies to:
\[ = \frac{(x_1 y_2 - y_1 x_2) \cdot \sqrt{(x_1^2 + y_1^2)(x_2^2 + y_2^2)}}{\sqrt{(x_1 y_2 - y_1 x_2)^2 + (x_1 x_2 + y_1 y_2)^2}} \]

### Final Result:
Thus, the final expression for \( OP_1 \cdot OP_2 \cdot \sin(\angle P_1OP_2) \) is:
\[ = x_1 y_2 - y_1 x_2 \]