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Parametric curve and normal

  1. Jun 5, 2010 #1
    1. The problem statement, all variables and given/known data

    Let [itex] c(t) = ( cos(At), sin(At), 1) [/itex] be a curve. (A is a constant)

    Show that the normal to [itex] c(t) [/itex] is always directed toward the z-axis.

    3. The attempt at a solution

    I am not sure how to show this. (For example, is the question "asking" us to show the cross product of something is 0 ?) If you tell me how to start the problem, I should have no problem.

    I have found the normal, which is [itex] N(t) = ( -cos(At), -sin(At), 0) [/itex].

    Thanks.
     
  2. jcsd
  3. Jun 5, 2010 #2

    rock.freak667

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    So you have N=<-cos(At),-sin(At),0> and this is in the form <x,y,z>.

    What is z equal to? What are the consequences of the negative sign in terms of direction?
     
  4. Jun 5, 2010 #3
    [itex] z = 1 [/itex] ? I mean [itex] z [/itex] is always equal to 1, unless you ask what [itex] z [/itex] is for the normal, which is 0. I'm not sure about your second question, could you explain more? Thanks.
     
  5. Jun 5, 2010 #4

    rock.freak667

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    I meant for the normal. If z=0, then you're normal is essentially in the xy-plane right?

    As for my other question if you have positive values of x and y, in relation to the z-axis, where would you plot those numbers? (Away or toward the axis when you keep increasing positively?)
     
  6. Jun 5, 2010 #5
    Away the z axis?
     
  7. Jun 5, 2010 #6

    rock.freak667

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    Right, so if you have <x,y,0> it points away from the z-axis. Where would <-x,-y,0> point?
     
  8. Jun 5, 2010 #7
    Directed toward the axis. I have a question though, cos(At) and sin(At) aren't always positive, so does this still work?

    Thanks.
     
  9. Jun 5, 2010 #8

    rock.freak667

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    I believe if you draw it out, you will see that when cosine is +ve, sine is -ve so one part of the normal will point towards the z-axis and when sine is +ve and cosine is -ve, the other part of the normal points towards the z-axis. In essence it will always point towards the z-axis.
     
  10. Jun 5, 2010 #9
    Thanks. How should we justify the answer though? I am not sure "what to say" to answer the question. Thanks.
     
  11. Jun 5, 2010 #10

    rock.freak667

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    Your normal is <-cos(At),-sin(At),0> or x= - cos(At), y= -sin(At), if you sketch this in the xy-plane you will get a circle. Each diameter will be a normal. As long as each one passes through the origin (where the z-axis would be perpendicular to the point (0,0)) that would illustrate it.

    The illustration would work I guess.
     
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