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Please prove this formula using the Pythagorean theorem

The formula in the image is the dot product formula for vectors \(\mathbf{v}_1\) and \(\mathbf{v}_2\), which is given by:

\[ x_1 x_2 + y_1 y_2 = \mathbf{v}_1 \cdot \mathbf{v}_2 = |\mathbf{v}_1| |\mathbf{v}_2| \cos \theta \]

To prove this formula using the Pythagorean theorem, we can follow these steps:

### Step 1: Express vectors in component form
Let \(\mathbf{v}_1\) and \(\mathbf{v}_2\) be two vectors in the plane with components:
\[ \mathbf{v}_1 = (x_1, y_1) \]
\[ \mathbf{v}_2 = (x_2, y_2) \]

### Step 2: Calculate the dot product of the vectors
The dot product of \(\mathbf{v}_1\) and \(\mathbf{v}_2\) is defined as:
\[ \mathbf{v}_1 \cdot \mathbf{v}_2 = x_1 x_2 + y_1 y_2 \]

### Step 3: Magnitude of the vectors
The magnitude (or length) of the vectors can be calculated using the Pythagorean theorem:
\[ |\mathbf{v}_1| = \sqrt{x_1^2 + y_1^2} \]
\[ |\mathbf{v}_2| = \sqrt{x_2^2 + y_2^2} \]

### Step 4: Angle between the vectors
Let \(\theta\) be the angle between the two vectors. Using trigonometric identities, we can express the dot product in terms of the magnitudes of the vectors and the cosine of the angle between them:
\[ \mathbf{v}_1 \cdot \mathbf{v}_2 = |\mathbf{v}_1| |\mathbf{v}_2| \cos \theta \]

### Step 5: Equate both expressions for the dot product
From Steps 2 and 4, we have two expressions for the dot product:
\[ x_1 x_2 + y_1 y_2 = |\mathbf{v}_1| |\mathbf{v}_2| \cos \theta \]

Thus, we have proved that:
\[ x_1 x_2 + y_1 y_2 = \mathbf{v}_1 \cdot \mathbf{v}_2 = |\mathbf{v}_1| |\mathbf{v}_2| \cos \theta \]

This completes the proof using the Pythagorean theorem and properties of vector magnitudes and angles.

Would you like further details or have any questions?

Here are five more questions you might want to explore:

1. How do you compute the cross product of two vectors?
2. Can you prove the Cauchy-Schwarz inequality using vectors?
3. What are some real-world applications of dot products?
4. How do you find the angle between two vectors given their coordinates?
5. What is the significance of the dot product being zero?

**Tip:** Understanding the geometric interpretation of vectors can greatly enhance your comprehension of vector operations and their applications.

Explore the proof of the dot product formula for vectors using the Pythagorean theorem. This solution details vector components, magnitudes, and the relationship between dot product and angle. Ideal for high school students studying advanced mathematics.

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