# Dot Product: Key Insights

## Numerical Definition

The dot product of two vectors is the sum of the products of their corresponding components:

\( \mathbf{v} \cdot \mathbf{w} = \sum_{i=1}^{n} v_i w_i \)

For example:

\( [1, 2] \cdot [3, 4] = 1 \cdot 3 + 2 \cdot 4 = 11 \)

## Geometric Interpretation

The dot product can be seen as the projection of one vector onto another:

\[\mathbf{v} \cdot \mathbf{w} = |\mathbf{v}| |\mathbf{w}| \cos(\theta)\]Where \( \theta \) is the angle between the vectors. Itâ€™s positive if they point in the same direction, zero if perpendicular, and negative if in opposite directions.

## Testing code blocks

```
# Python example code
def hello_world():
print("Hello, world!")
```