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Linear dependence/independence

  1. Jun 6, 2006 #1
    Given that and u||v, v is not parallel with w and z= −3u state (with reasons) whether the following vectors are linearly dependent with u.

    a) -2v -> LD since u is parallel with v, u must be also parallel with 2v

    b) 3w -> LI since u is parallel with v and v isn't parallel with w, w is also not parallel with u

    c) 4z -> LD since u is parallel with v and v is parallel with z (z is just a scalar mutliple of u anyways), z must be parallel with u

    d) 3v + 4z -> LD since v and z are parallel with each other and both are parallel with u, any combination of v and z is parallel with u

    can someone check these for me? the coefficients infront of the vectors have no real siginificance right?, there just scalar multiples? if they do, then i screwed up big time.
     
  2. jcsd
  3. Jun 7, 2006 #2
    different topic (division of a line) but i don't want to pollute the board with new threads.

    1. State the ratio into which A divides BC. Indicate whether the division is internal or external.

    a) BA = 2/3(BC)

    is this question worded right? because you get BA/BC = 2/3 which really means B divides AC externally in the ratio 2:3. also, usually if C divides AB externally, the ratio would be AC:CB, which means one of them is negative. that's the second problem with this question even if the wording is incorrect.
     
  4. Jun 7, 2006 #3
    For the first question, I don't find anything wrong with your answers.
    Remember that a scalar coefficient in front of a vector, simply scales the vector by some measure while maintaining its direction .

    For the second question, if you are not talking of vectors, but merely scalars or line segments, both the answers are possible . However if it is vectors the situation changes and there is only one answer .
    Before I explain, could you please clarify further .
     
  5. Jun 7, 2006 #4

    HallsofIvy

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    I don't understand why you are worrying about "B divides AC externally" or "C divides AB externally". The problem asked in what ratio A divides BC. Clearly, A is 2/3 of the way from B to C and so divides BC 2 to 1. (BA is twice as long as AC)
     
    Last edited: Jun 7, 2006
  6. Jun 7, 2006 #5
    HallsofIvy: yes thanks for clarifying, i needed to draw a diagram to visualize.

    B______________A______C

    it would look something like that i suppose.
     
  7. Jun 7, 2006 #6
    It can also be something like

    A________B_______________C

    Note that in this case too BA = 2/3 BC and division is external and in which case AB/BC = (2/3)/(1+2/3) = 2/5 .Thus both internal and external division are possible as I said in my earlier post.
    Now had it been
    [tex]\vec{BA} = \frac{2}{3}\vec{BC}[/tex]
    the situation would have been different .
    Can you see how ? ;)
     
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