Dot Product of a Unit Vector with the Negative of itself

[EarthDecon]

Homework Statement

I am trying to calculate the flux for the octant of a sphere, and I am trying to figure out how the mathematics, dot products, and dA works in the integral. I already did the quadrant for \hat{θ} where θ= π/2 (the bottom quadrant) and I did the left quadrant where \hat{n} = -\hat{ø} but I want to understand, when you are calculating \int_{S}\vec{F}\bullet\hat{ø}dA , for F_{ø}, if ø is negative for the left quadrant, I know that the flux will be 0 because ø = 0 and sinø = 0 but in the integral, will the negative cancel out?

Homework Equations

Here is a picture for the octant of the sphere, my question is on the left quadrant: http://2.bp.blogspot.com/_N2c1FjhWLag/TF_yaJpc6DI/AAAAAAAAACI/4hhPUla483A/s1600/sphel.gif

To solve for the total flux we have the equation:
\oint \vec{F}\bullet\hat{n}dA = \int_{S1}\vec{F}\bullet\hat{n}dA + \int_{S2}\vec{F}\bullet\hat{n}dA + \int_{S3}\vec{F}\bullet\hat{n}dA + \int_{S4}\vec{F}\bullet\hat{n}dA

\vec{F} (r,θ,ø) = (r^{2}cosθ)\hat{r}+(r^{2}cosø)\hat{θ}-(r^{2}cosθsinø)\hat{ø}

The Attempt at a Solution

I already know that the flux is equal to zero as I said, I just wanted to be sure that the dot product of two unit vectors in this case is -1 or if there's more calculation needed for this.

 

[Simon Bridge]

Why not check by explicitly taking a unit vector $\hat u = (1,0)$ and formally constructing the dot product of that with the negative of itself?