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Math Help - Vectors

  1. #1
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    Vectors

    Hello!

    Q: With respect to an origin O, the points P and Q have position vectors p and q respectively, given by p = (cos t)i + (sin t)j - k, q = (cos 2t)i - (sin 2t)j + \frac{1}{2}k, where t is a real parameter such that 0 < t < 2\pi. Given that p  \cdot q = cos 3t - \frac{1}{2}, hence or otherwise, find the greatest value of angle POQ.




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  2. #2
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    The dot product of two vectors is given by
    |\vec{u}||\vec{v}| cos(\theta) where /theta is the angle between the vectors.
    |p|= \sqrt{cos^2(t)+ sin^2(t)+ 1}= \sqrt{2} while |q|= [tex]\sqrt{cos^2(2t)+ sin^2(2t)+ 1/4}= \sqrt{2}{2}.

    Since p \cdotq= cos(3t)- 1, we must have
    cos(3t)-1 = cos(\theta) so the problem is simply "what is the maximum value of cos(3t)-1 for t between 0 and [tex]2\pi[/itex]" and then take the arccosine of that. Since cosine ranges between -1 and 1, cos(3t)-1 ranges between -2 and 0. -1, the smallest possible value of cosine, is included in that and so the largest possible angle between p and q is simply the largest possible angle between any two vectors, 180 degrees.
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  3. #3
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    Thank you for helping!

    May I know what did you evaluate |q| to be? I thought |q| = \sqrt{5/4}?
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