If O be the origin and `Q_(1)(x_(1),y_(1))` and `Q_(2)(x_(2),y_(2))` be two points then `OQ_(1).OQ_(2)cos (/_Q_(1)OQ_(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 \( OQ_1 \cdot OQ_2 \cdot \cos(\angle Q_1 O Q_2) \). Here are the steps to derive the required expression:

### Step 1: Identify the distances
Let \( O \) be the origin \( (0, 0) \), \( Q_1 \) be the point \( (x_1, y_1) \), and \( Q_2 \) be the point \( (x_2, y_2) \).

The distance \( OQ_1 \) can be calculated using the distance formula:
\[ OQ_1 = \sqrt{x_1^2 + y_1^2} \]
Similarly, the distance \( OQ_2 \) is:
\[ OQ_2 = \sqrt{x_2^2 + y_2^2} \]

### Step 2: Calculate the distance between points \( Q_1 \) and \( Q_2 \)
The distance \( Q_1Q_2 \) is given by:
\[ Q_1Q_2 = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step 3: Use the cosine rule
According to the cosine rule in triangle \( OQ_1Q_2 \):
\[ OQ_1^2 + OQ_2^2 - 2 \cdot OQ_1 \cdot OQ_2 \cdot \cos(\angle Q_1 O Q_2) = Q_1Q_2^2 \]
Rearranging gives us:
\[ OQ_1 \cdot OQ_2 \cdot \cos(\angle Q_1 O Q_2) = \frac{OQ_1^2 + OQ_2^2 - Q_1Q_2^2}{2} \]

### Step 4: Substitute the distances into the equation
Substituting the distances we calculated:
\[ OQ_1^2 = x_1^2 + y_1^2 \]
\[ OQ_2^2 = x_2^2 + y_2^2 \]
\[ Q_1Q_2^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 \]

Expanding \( Q_1Q_2^2 \):
\[ Q_1Q_2^2 = (x_2^2 - 2x_1x_2 + x_1^2) + (y_2^2 - 2y_1y_2 + y_1^2) \]
\[ = x_1^2 + y_1^2 + x_2^2 + y_2^2 - 2(x_1x_2 + y_1y_2) \]

### Step 5: Substitute back into the cosine rule equation
Now substituting back into the rearranged cosine rule equation:
\[ OQ_1 \cdot OQ_2 \cdot \cos(\angle Q_1 O Q_2) = \frac{(x_1^2 + y_1^2) + (x_2^2 + y_2^2) - (x_1^2 + y_1^2 + x_2^2 + y_2^2 - 2(x_1x_2 + y_1y_2))}{2} \]
This simplifies to:
\[ OQ_1 \cdot OQ_2 \cdot \cos(\angle Q_1 O Q_2) = \frac{2(x_1x_2 + y_1y_2)}{2} = x_1x_2 + y_1y_2 \]

### Final Result
Thus, we conclude that:
\[ OQ_1 \cdot OQ_2 \cdot \cos(\angle Q_1 O Q_2) = x_1x_2 + y_1y_2 \]