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:

Where \( \theta \) is the angle between the vectors. It’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!")