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?