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Math Help - Vector Functions

  1. #1
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    Vector Functions

    Question 1:
    Two particles travel along the space curves
    r1(t)=<t, t2, t3>, r2(t)=<1+2t, 1+6t, 1+14t>
    Do the particles collide? Do their paths intersect?

    I know how to find out how the particles collide, just set the corresponding components of both vectors equal to one another and look for a t-value that works for all of them (which there isn't). My question is how do you determine if their paths intersect?

    Question 2:
    Use the parametric equations
    x=(2+cos(1.5t))cos(t)
    y=(2+cos(1.5t))sin(t)
    z=sin(1.5t)
    to sketch the curve by hand as viewed from above, with gaps indicating where the curve passes over itself. Start by showing that the projection of the curve onto the xy-plane has polar coordinates r=2+cos(1.5t) and Θ=t, so r varies between 1 and 3. Then show that z has maximum and minimum values when the projection is halfway between r=1 and r=3.
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  2. #2
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    Quote Originally Posted by desperatestudent View Post
    Question 1:
    Two particles travel along the space curves
    r1(t)=<t, t2, t3>, r2(t)=<1+2t, 1+6t, 1+14t>
    Do the particles collide? Do their paths intersect?

    I know how to find out how the particles collide, just set the corresponding components of both vectors equal to one another and look for a t-value that works for all of them (which there isn't). My question is how do you determine if their paths intersect?

    [snip]
    Assume that there's an intersection point. Let particle A be at that point at t = t_A and particle B be at the point at t = t_B. Solve the following equations simultaneously for t_A and t_B:

    t_A = 1 + 2 t_B .... (1)

    t_A^2 = 1 + 6 t_B .... (2)

    t_A^3 = 1 + 14 t_B .... (3)

    I suggest solving (1) and (2) simultaneously and then testing the solutions in the third. If you get a solution that works for all three equations, then those values can be used to get the cartesian coordinates of the intersection point.
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  3. #3
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    thx mr. fantastic!

    also, i realize for my second question that it would be rather difficult to do what the question is asking on a forum, so I'll explain my biggest problem with the question. When it says "graph from above" does it mean I should basically ignore z in my graph since it's technically coming out of the page at me?
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  4. #4
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    Quote Originally Posted by desperatestudent View Post
    thx mr. fantastic!

    also, i realize for my second question that it would be rather difficult to do what the question is asking on a forum, so I'll explain my biggest problem with the question. When it says "graph from above" does it mean I should basically ignore z in my graph since it's technically coming out of the page at me?
    That is what I would do.
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  5. #5
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    hm ok
    what puzzles me though, is that the question asks me to leave blanks where the curve goes over itself. Yet, after graphing the polar equation, I can't identify where these are or how I find them :\
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