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Math Help - orthogonal vectors

  1. #1
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    orthogonal vectors

    I don't understand how to solve this problem, help please.

    The Vectors  U = cos\theta I+sin\theta J, V = -sin\theta I+cos\theta J for any  \theta , form an orthonormal basis for the plane; they are orthogonal vector of length 1. Let \theta = \frac{2\pi}{4} and  x = 5I+8J. Then we can write  X = uU + vV

    with u = ___, v = ___
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  2. #2
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    Quote Originally Posted by viet View Post
    I don't understand how to solve this problem, help please.
    Substitute \theta = \frac{2 \pi}{4} = \frac{\pi}{2} into both U\; \text{and} \; V giving U = \cos \frac{\pi}{2} i + \sin \frac{\pi}{2}\, j \; = j \;\;\text{and} \; V = - \sin \frac{\pi}{2}\, i + \cos \frac{\pi}{2} j = - i. So X = u U + v V = u j + v (-i) = 5 i + 8 j. Now compare.
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