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Homework Help: Linear independence and vector parametrics!

  1. Jan 25, 2010 #1
    1. The problem statement, all variables and given/known data
    Consider the two lines, given in the paramentric form

    L1: x = (0, 1 ,2) + s(1, 0, 2)
    L2: x = (4, 2, c) + t(-2, 0, d)

    where c and d are constants.

    a) For what value of d are the lines parallel?
    b) With the value of d above, for what value(s) of c (if any) are the two lines identical? Justify briefly.
    c) For the case c = 5 and d = 0 find the point P on L1 and Q on L2 such that the distance between P and Q is as small as possible.

    2. A concept question:
    Can any vector x with two components be expressed as a linear combination of a and b? Why?
    2. Relevant equations

    ?

    3. The attempt at a solution
    a) I think d = -4, but not sure if that's right.
    b) I'm not sure how to approach this problem. Do I make 2 + 2s = c - 4t
    But what do I do after that?
    c) No clue. I'm thinking perhaps using the dot product = 0?

    2. I think a vector can only be expressed as a linear combination of a and b if the two vectors are linearly independent. How do I prove that?
     
  2. jcsd
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