### Instruction: The set of vectors $\mathbf{v}$ such that
\[\mathbf{v} \cdot \mathbf{v} = \mathbf{v} \cdot \begin{pmatrix} 2 \\ 0 \end{pmatrix}\]form a curve in the plane.  Find the area of the region contained in the curve.### Response: I want to find the area of the region bounded by the curve, so I need to know what the curve looks like.
To do that, I can rewrite the given equation in terms of the components of $\mathbf{v}$.
Let $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$.
Then the equation becomes \[x^2 + y^2 = 2x.\]
This looks like a quadratic equation in $x$, but I can also see that it is the equation of a circle with center $(1,0)$ and radius $1$.
To confirm this, I can complete the square for $x$ and get \[(x-1)^2 + y^2 = 1.\]
So the curve is a circle, and the region contained in it is a circular disk.
The area of a circular disk is given by the formula $\pi r^2$, where $r$ is the radius.
In this case, $r = 1$, so the area is just $\pi$.