Dot Product (in R^2 and R^n)

There does indeed exist a notion of multiplying two vectors together, though it's slightly different from the multiplication that we're used to seeing.

The dot product (or scalar product) of two vectors $\vec{v}$ and $\vec{w}$ is defined as:

$$\vec{v} \cdot \vec{w} = |\vec{v}| |\vec{w}| \cos\theta$$

where $\theta$ is the angle (in degrees or radians) made between the two vectors when both of their "tails" are placed at the origin, i.e.

You might be asking yourself: why is this seemingly arbitrary definition a notion of multiplication?

Why not multiply the two vectors component by component, similar to how addition, subtraction, and scal

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