If O be the origin and `A(x_(1), y_(1)), B(x_(2), y_(2))` are two points, then what is `(OA) (OB) cos angle AOB` ?

Options

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

Explanation

To solve the problem, we need to find the expression for \( (OA)(OB) \cos \angle AOB \) given the points \( O(0, 0) \), \( A(x_1, y_1) \), and \( B(x_2, y_2) \).

### Step-by-Step Solution:

1. **Find the distances \( OA \) and \( OB \)**:
- The distance \( OA \) from the origin \( O \) to point \( A \) is given by:
\[ OA = \sqrt{(x_1 - 0)^2 + (y_1 - 0)^2} = \sqrt{x_1^2 + y_1^2} \]
- The distance \( OB \) from the origin \( O \) to point \( B \) is given by:
\[ OB = \sqrt{(x_2 - 0)^2 + (y_2 - 0)^2} = \sqrt{x_2^2 + y_2^2} \]

2. **Find the distance \( AB \)**:
- The distance \( AB \) between points \( A \) and \( B \) is given by:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

3. **Use the cosine rule to find \( \cos \angle AOB \)**:
- According to the cosine rule:
\[ \cos \angle AOB = \frac{OA^2 + OB^2 - AB^2}{2 \cdot OA \cdot OB} \]
- Substituting the values we found:
\[ \cos \angle AOB = \frac{(x_1^2 + y_1^2) + (x_2^2 + y_2^2) - ((x_2 - x_1)^2 + (y_2 - y_1)^2)}{2 \cdot OA \cdot OB} \]

4. **Simplify the expression**:
- Expanding \( (x_2 - x_1)^2 + (y_2 - y_1)^2 \):
\[ (x_2 - x_1)^2 + (y_2 - y_1)^2 = x_2^2 - 2x_1x_2 + x_1^2 + y_2^2 - 2y_1y_2 + y_1^2 \]
- Thus, substituting back into the cosine formula:
\[ \cos \angle AOB = \frac{x_1^2 + y_1^2 + x_2^2 + y_2^2 - (x_2^2 - 2x_1x_2 + x_1^2 + y_2^2 - 2y_1y_2 + y_1^2)}{2 \cdot OA \cdot OB} \]
- After cancellation, we get:
\[ \cos \angle AOB = \frac{2x_1x_2 + 2y_1y_2}{2 \cdot OA \cdot OB} \]

5. **Calculate \( (OA)(OB) \cos \angle AOB \)**:
- Now, we can find \( (OA)(OB) \cos \angle AOB \):
\[ (OA)(OB) \cos \angle AOB = OA \cdot OB \cdot \frac{2x_1x_2 + 2y_1y_2}{2 \cdot OA \cdot OB} \]
- The \( OA \cdot OB \) terms cancel out:
\[ (OA)(OB) \cos \angle AOB = x_1x_2 + y_1y_2 \]

### Final Result:
Thus, the final result is:
\[ (OA)(OB) \cos \angle AOB = x_1x_2 + y_1y_2 \]