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Orthogonal vectors in R4?

  1. Mar 25, 2014 #1
    1. The problem statement, all variables and given/known data
    Given following vectors in R4:
    v= (4,-9,-6,3)
    u = (5,-8,k,4)
    w=(s,-5,4,t)

    A. Find value of k if u and v are orthogonal
    B. Find values of S and T if w and u are orthogonal and w and v are orthogonal


    2. Relevant equations

    Orthogonal means dot product is zero



    3. The attempt at a solution

    A. I already solved and got k= 104/6 which is correct


    B.
    I tried to solve this one assuming that k was still equal to 104/6 and got
    s=4 and t=-37/3
    But this is not the answer.
    Do I assume that k is still the same result as in part A or can k be anything?
    If k is anything than I get:

    [
    4 3 -21
    5 4 (-40-4k)
    ]
    as a matrix, but how would I solve this?
     
  2. jcsd
  3. Mar 25, 2014 #2

    jbunniii

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    I think you should continue to use ##k=104/6## in part B. Can you show how you calculated ##s## and ##t##? You can easily check that with those values, ##w## is orthogonal to ##v## but not to ##u##.
     
  4. Mar 25, 2014 #3
    Okay thats what I thought b/c if k is unknown you have 3 variables which wouldn't be solveable.

    So if k= 104/6
    v*w = 4s + 45 - 24 + 3t = 0
    u*w = 5s + 40 + 416/6 + 4t = 0

    From there I combined like terms:

    4s + 3t = -21
    5s + 4t = -656/6

    Then I formed a matrix and use row operations to find that:

    s = 244
    t = -997/3

    Is this right? I think my mistake the first time using this method was when I got the 40 and 416/6 on the right hand side I added 40 instead of subtracting which changed the value.
     
  5. Mar 25, 2014 #4

    jbunniii

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    Yes, that looks right. You can always substitute your answers back into your expressions for v*w and u*w to verify that you get zero in both cases.
     
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