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Orthogonal vectors

  1. Nov 21, 2012 #1
    How do you check if 2 vectors are orthogonal?

    I know that if 2 vectors are orthogonal, then there dot product is 0. But I don't think that necessarily means if their dot product is 0, the 2 vectors are orthogonal. Like what if you had 2 zero vectors, their dot produt would be 0, but they're not orthogonal.

    I also know that the angle between the 2 vectors is 90 degrees. I think this one that 2 vectors that are 90 degrees apart are orthogonal. Right?
     
  2. jcsd
  3. Nov 21, 2012 #2

    Mark44

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    Sure they are. The zero vector is considered to be orthogonal to every other vector, including another zero vector.
    Right.
     
  4. Nov 21, 2012 #3
    Okay thank you.

    Also if we have 2 vectors u and v, and vector w is the projection of u onto v.
    How is the length of w determined?
     
  5. Nov 21, 2012 #4

    haruspex

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    It depends on the projection. If you mean an orthogonal projection (i.e. orthogonal to v) then it will satisfy (u-w).v = 0
     
  6. Nov 21, 2012 #5
    Uh, for this problem I'm doing the instructions are "The vector w is called the orthogonal projection of u
    onto v. Sketch the three vectors u, v, and w."

    I attached my work.
    Vector u is given to be [-2 3]
    v is given to be [4 0]


    I calculated the orthogonal projection of u onto v
    by 1st finding the dot product of u and v.
    Then dividing that by the magnitude of vector v squared.
    Then multipling that by vector v.
    This gave:
    w=[-2 0]
     

    Attached Files:

  7. Nov 21, 2012 #6

    haruspex

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    Looks like a valid method and the right answer. And it does satisfy (u-w).v = 0.
     
  8. Nov 21, 2012 #7
    I think for this one you can find out by inspection since vector v is on the x axis. I jsut saw that the projection equation I used is introduced in a later section, so I probably should have used another method.
    What other method is there if this were not an obvious case where v is not on the x axis? As in how do you find the length of vector w.
     
  9. Nov 21, 2012 #8

    haruspex

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    I don't see an easier way than deriving that equation.
    It's clear that w = λv for some scalar λ. And orthogonality gives (u-w).v = 0, right?
    Substitute for w and determine λ.
     
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