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Dot Product of a Unit Vector with the Negative of itself

  1. Apr 14, 2014 #1
    1. The problem statement, all variables and given/known data

    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 [itex]\hat{θ}[/itex] where θ= π/2 (the bottom quadrant) and I did the left quadrant where [itex]\hat{n}[/itex] = -[itex]\hat{ø}[/itex] but I want to understand, when you are calculating [itex]\int_{S}[/itex][itex]\vec{F}[/itex][itex]\bullet[/itex][itex]\hat{ø}[/itex]dA , for F[itex]_{ø}[/itex], 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?

    2. Relevant 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:
    [itex]\oint[/itex] [itex]\vec{F}[/itex][itex]\bullet[/itex][itex]\hat{n}[/itex]dA = [itex]\int[/itex][itex]_{S1}[/itex][itex]\vec{F}[/itex][itex]\bullet[/itex][itex]\hat{n}[/itex]dA + [itex]\int[/itex][itex]_{S2}[/itex][itex]\vec{F}[/itex][itex]\bullet[/itex][itex]\hat{n}[/itex]dA + [itex]\int[/itex][itex]_{S3}[/itex][itex]\vec{F}[/itex][itex]\bullet[/itex][itex]\hat{n}[/itex]dA + [itex]\int[/itex][itex]_{S4}[/itex][itex]\vec{F}[/itex][itex]\bullet[/itex][itex]\hat{n}[/itex]dA

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

    3. 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.
  2. jcsd
  3. Apr 15, 2014 #2

    Simon Bridge

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    Science Advisor
    Homework Helper

    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?
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