I am trying to calculate a line integral around a triangle that is oriented counter-clockwise.
The triangle is bounded by: $$y=0$$ $$x=3$$ $$y=x$$
I know how to parameterize the y=0 and x=3 parts of the triangle but I am unsure how to parameterize the y=x part as it is oriented counter clockwise.
For $c_1$ and $c_2$ I get $$\vec{r_1}=<t,0>$$ and $$\vec{r_2}=<3,t>$$
How can I get $\vec{r_3}$?
Given two points $\mathbf p$ and $\mathbf q$, a simple parameterization of the line segment joining them is $\mathbf r(t) = (1-t)\mathbf p+t\mathbf q$. You can see that $\mathbf r(0)=\mathbf p$ and $\mathbf r(1)=\mathbf q$. Choose the start and end points of your line segment accordingly to get the correct orientation for it.