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Math Help - Vector Calculus problems, where to start?

  1. #1
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    Vector Calculus problems, where to start?

    (1) Find an equation of a plane, II, such that is has the following properties:
    (a) The minimum distance from the point (7, -3, 2) to II is equal to the minimum distance from the point (7, 3, -1) to II.
    (b) The plane, II, does not contain either of the points (7, -3, 2) or (7, 3, -1).
    (c) The plane, II, contains the point (12, 1, 2.5).
    (d) The plane, II, is not parallel to the yz-plane.

    (2) The curve defined by the vector valued function, r(t) = {4 cos(t), sqrt(32) sin(t), 4 cos(2t)}, can be viewed as "living", or existing, on a quadratic surface. What is an equation of a quadratic surface that r(t) lives on? What is the name of this type of quadratic surface?

    (3) Show that the plane ax + by + cz = d and the line r(t) = r + tv, not in the plane, have no points of intersection if and only if v (dot product) {a,b,c}=0. Give a geometric explanation of the result.

    I don't expect you guys/gals to do all my work but I'm having trouble finding a starting approach to any of these problems. Any hints would be greatly appreciated. Thank you so much!
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  2. #2
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    Re: Vector Calculus problems, where to start?

    In (1), if the plane contains the line x=7, (y,z) = (0,1/2) + (3,6)t, it will meet (a) based on the geometry (my line is a perpendicular bisector of the line segment between the two points). Then if it meets (c), it'll automatically meet (b) and (d). So find the plane containing the line and the point.

    For (2), you use trig identities to find a relationship between x, y, and z. The identity \cos{2t}=\cos^2{t}-\sin^2{t} comes to mind.

    For (3), {a,b,c} has a geometrical meaning..., and the dot product being zero has a geometrical meaning....

    - Hollywood
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