2D Rotation

I've got a vector, which represents the direction an agent is "facing" within a maze's frame of reference.  OpenGL will allow me to draw the agent's rotation by rotating its own frame of reference--but the agent's "facing" vector remains unchanged within the agent's frame of reference.  I am unable to determine the vector's value in the maze's frame of reference from OpenGL.

What I need, then, is to do the rotation myself, and pass the result into OpenGL (without using OpenGL to rotate the agent's frame of reference).  In this case, I will be rotating a single vector value around the z-axis, so it looks just like a two-dimensional rotation.

If I rotate the vector represented by the red line by θ (theta), the result will be the vector represented by the green line.  α (alpha), is the angle between the x-axis and the vector.  We'll call the original vector V₀ (with components x₀, y₀), and the vector resulting from the rotation V₁ (with x₁, y₁).  The length of the vector is r.

x₀ and y₀ are found by simple trigonometric functions on α:

x₀ = r∙cos(α)
y₀ = r∙sin(α)

Likewise, x₁ and y₁ are found in terms of α and θ:

x₁ = r∙cos(α + θ)
y₁ = r∙sin(α + θ)

 Since cos(a+b) = cos(a)∙cos(b) - sin(a)∙sin(b), and sin(a+b) = sin(a)∙cos(b) + cos(a)∙sin(b):

x₁ = r( cos(α)∙cos(θ) - sin(α)∙sin(θ) )
x₁ = r∙cos(α)∙cos(θ) - r∙sin(α)∙sin(θ)

x₁ = x₀∙cos(θ) - y₀∙sin(θ)

y₁ = r( sin(α)∙cos(θ) + cos(α)∙sin(θ) )
y₁ = r∙sin(α)∙cos(θ) + r∙cos(α)∙sin(θ)
y₁ = y₀∙cos(θ) + x₀∙sin(θ)

y₁ = x₀∙sin(θ) + y₀∙cos(θ)

These equations can be represented in matrix form, but for now they suffice to rotate the agent's "facing" vector.