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Unit Vector geometry

  1. Feb 5, 2012 #1
    1. The problem statement, all variables and given/known data
    If u is a unit vector, find u.v and u.w


    2. Relevant equations
    I assumed that unit vector means u=<1,1,1>
    u.v=|u||v|cos60
    My knowledge of unit vectors is very limited. I know that a unit vector is
    i=<1,0,0>
    j=<0,1,0>
    k=<0,0,1>

    3. The attempt at a solution
    Since the triangle is isoceles. |u|=|v|
    u.v=(sqrt(3)+sqrt(3) )cos60=-3.29925

    Is this correct?
     
    Last edited: Feb 5, 2012
  2. jcsd
  3. Feb 5, 2012 #2
    An image of the above problem has been attached.
     

    Attached Files:

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  4. Feb 6, 2012 #3
    Can someone answer?
     
  5. Feb 6, 2012 #4

    Mark44

    Staff: Mentor

    No, this is not a unit vector. It's not a unit vector because its length is not 1. (Its length is √3.)
    Yes, these are three unit vectors, but a unit vector can point in any direction, not just in the direction of the axes.
     
    Last edited: Feb 6, 2012
  6. Feb 6, 2012 #5
    I thought that the vectors shown in my graph could be considered as forming a triangle.

    The graph is attached, after the first post.
     
  7. Feb 6, 2012 #6

    Mark44

    Staff: Mentor

    I'm confused. Your drawing shows tickmarks on all three sides, which implies that all three are equal in length and that the triangel is equilateral. In post #1, you say that the triangle is isosceles, which implies that only two sides are equal. Which is it?
     
  8. Feb 6, 2012 #7
    I'm sorry, it's an equilateral triangle. The graph is correct. The problem does not state anything other than what is shown in the graph and the first post.
     
    Last edited: Feb 6, 2012
  9. Feb 6, 2012 #8

    Mark44

    Staff: Mentor

    Your sketch is NOT correct if the triangle is only isosceles. The tickmarks on all thee sides imply that it is equilateral, which is different from being isosceles.

    Are you given that all three sides are equal in length? If so, then stating that the triangle is isosceles confuses the matter.
     
  10. Feb 6, 2012 #9
    You are right, it's equilateral.
     
  11. Feb 6, 2012 #10

    Mark44

    Staff: Mentor

    OK, we're finally getting somewhere. If u is a unit vector, then so are v and w.

    Another formulation for the dot product is u[itex]\cdot[/itex]v = |u||v|cosθ. In this case, θ is 60°, as are the other two interior angles.

    So what do you get for u[itex]\cdot[/itex]v and u[itex]\cdot[/itex]w?
     
  12. Feb 6, 2012 #11
    u.v= (u^2)cos60=u^2(1/2)

    u.w=(u^2)cos60=(u^2)1/2
     
  13. Feb 6, 2012 #12

    Mark44

    Staff: Mentor

    No. Do you understand what the term unit vector means?

    Also, what does u2 even mean for a vector?
     
  14. Feb 7, 2012 #13
    I wrote u^2, but I meant |u^2|. I realize that there's no multiplication of vectors.

    I don't think I know, what a unit vector means. I know that a unit vector is (vector/|vector|), but that's it.
     
  15. Feb 7, 2012 #14

    Mark44

    Staff: Mentor

    That still doesn't make any sense, because you are multiplying a vector by itself, and then taking the magnitude of the result.

    This makes more sense: |u|2, the square of the magnitude of vector u.
    If there's something you don't know, your very first impulse should be to find out how it is defined. Math books are careful to define the terms they use very precisely. A unit vector is any vector whose length is 1. That's why they call it a unit vector - the word "unit" implies "oneness".
     
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