# Math Help - Line Integral of a vector field

1. ## Line Integral of a vector field

Let C be the triangle with vertices (-1,1),(1,-1),(0,2) traversed anticlockwise

Find \int _C v dr where v(x,y) = (y,-x)

I parameterised the triangle by splitting it into three lines c_1(t)=(1-t,3t-1), c_2(t)=(-t,2+3t), c_3(t)=(2t-1,-1) all with t in [0,1]. This gives three integrals, and plugging it into the formula in the definition: \int_C F(r)dr = \int_a^b F(r(t)).r'(t) dt
I got the answer as -6.

Firstly is this the right answer? Secondly the negative sign is simply because we traversed anticlockwise?

2. If you take anticlockwise orientation, one of the lines is the segment from A (-1 , 1) to B (1 , -1) :

$\gamma(t)=A+\overrightarrow{AB}=(-1+2t,1-2t)\quad (t\in[0,1])$

Where is this line?