Overview
The dot product is the standard inner product that is used in {% \mathbb{R}^n %}. Given a vector defined as an ordered set of numbers
{% \vec{x} = (x_1,x_2, ... ,x_n) %}
the dot product is defined to be:
{% \vec{x} \cdot \vec{y} = \sum_i x_i y_i %}
Geometric Interpretation
{% \vec{x} \cdot \vec{y} = |\vec{x}| |\vec{y}| cos(\theta) %}
Projection
The projection of the vector {% \vec{y} %} onto the vector {% \vec{x} %}, where projection is defined as a vector pointing along the {% \vec{x} %} direction but scaled by the length of parallel component of {% y %} along the {% \vec{x} %} direction is given by:
{% (\vec{x} \cdot \vec{y}) \vec{x} %}
where {% \vec{y} %} is a vector of unit length.