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Math Help - Unit Vector Question

  1. #1
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    Unit Vector Question

    Going back over textbook problems and having a bit of trouble with this one.
    If v=ai+bj, show that a/((a^2+b^2)^1/2)=Cos theta & b/((a^2+b^2)^1/2)=Sin theta, where theta is the direction of v.
    Sorry if the question is not written the best. Newbie to the site. Thanks for your help in advance.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Joeda View Post
    Going back over textbook problems and having a bit of trouble with this one.
    If v=ai+bj, show that a/((a^2+b^2)^1/2)=Cos theta & b/((a^2+b^2)^1/2)=Sin theta, where theta is the direction of v.
    Sorry if the question is not written the best. Newbie to the site. Thanks for your help in advance.
    I would suggest that you first draw a diagram.

    The vector \mathbf{v}=a\mathbf{i}+b\mathbf{j} has a component of length a in the x direction, and a component of length b in the y direction. The length of the hypotenuse of the triangle (which is the length of the vector) is given by \parallel\!\mathbf{v}\!\parallel=\sqrt{a^2+b^2}.

    Now, assign the angle between the vector and the positive x axis the value \theta. Can you try to visualize what I described?

    You should then be able to determine \cos\theta and \sin\theta.

    Does this make sense?
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  3. #3
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    That does make sense thanks.
    So to answer the question would I just have to draw the diagram and label all vectors?
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Joeda View Post
    That does make sense thanks.
    So to answer the question would I just have to draw the diagram and label all vectors?
    That isn't the answer itself. It leads you to the answer, which is to show that \cos\theta=\frac{a}{\sqrt{a^2+b^2}} and \sin\theta=\frac{b}{\sqrt{a^2+b^2}}
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  5. #5
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    ok so i think ive got it. Im using {v} as my symbol for magnitude of v.

    v=ai+bj
    {v}=((a^2+b^2)^1/2)

    and u=v/{v}
    =a/((a^2+b^2)^1/2)

    and we have this equatioin: u=(cos theta)i + (sin theta)j

    so how do I put it all together to prove the question?
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  6. #6
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Joeda View Post
    ok so i think ive got it. Im using {v} as my symbol for magnitude of v.

    v=ai+bj
    {v}=((a^2+b^2)^1/2)

    and u=v/{v}
    =(ai+bj)/((a^2+b^2)^1/2)

    and we have this equatioin: u=(cos theta)i + (sin theta)j

    so how do I put it all together to prove the question?
    I inserted missing information in your post in red.

    Now break it up into \frac{a}{\sqrt{a^2+b^2}}\mathbf{i}+\frac{b}{\sqrt{  a^2+b^2}}\mathbf{j} and then the result follows.
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