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Polar Coordinate Inner Product

  1. Sep 30, 2012 #1
    Just starting up school again and having trouble remembering some mathematics. Here's the problem.

    Find the inner product of ⃗a = (1, 45◦) and ⃗b = (2, 90◦), where these vectors are in polar coordinates (r, θ).

    Thanks =) 1st post here btw.
     
  2. jcsd
  3. Oct 7, 2012 #2
    http://www.iancgbell.clara.net/maths/vectors.htm

    About midway down this page you can see that the dot product in polar coordinates is [itex]\small (r_1,\theta_1) \tiny \bullet \small (r_2,\theta_2) = r_1r_2 \cos(\theta_1-\theta_2)[/itex]. One solution is to use this formula.

    The other one is just to represent the vectors in normal cartesian coordinates as [itex]\frac{1}{\sqrt{2}}(1,1)[/itex] (the factor 1/sqrt(2) is there to make the vector have length 1) and [itex](2,0)[/itex] and then use the normal dot product.
     
    Last edited by a moderator: Dec 27, 2013
  4. Oct 7, 2012 #3

    HallsofIvy

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    This is more calculus than "linear algebra". There are two ways to go:
    1) The "non-coordinate" definition of the dot product: [itex]u\cdot v= |u||v|cos(\theta)[/itex], where [itex]\theta[/itex] is the angle between the two vectors. Here, |u|= 1, |v|= 2 and angle between them is 45 degrees.

    2) Convert to Cartesian coordinates. The vector at 45 degrees with length 1 is [itex]<\sqrt{2}/2, \sqrt{2}/2>[/itex]. The vector at 90 degrees, with length 2, is <0, 1>.
     
    Last edited by a moderator: Dec 27, 2013
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