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Math Help - Question regarding Vectors

  1. #1
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    Question regarding Vectors

    Hi All!

    I have this nagging problem that I've been at it for a long time. I have attached a scanned sheet with the question. I really hope someone can help me out or atleast push me in the right direction.

    Thanks a bunch in advance!

    Akshay
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  2. #2
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    One difficulty is that it is only the point E that is height "l" above B. D, for example, cannot be height "l" above A without "warping" the plane. But we can calculate the angle. By the Pythagorean theorem, the distance from O to B is \sqrt{a^2+ c^2} so the angle between plane 0ABC and plane ODEF (and so both all three \alpha, \beta, and \gamma) are arctan\left(\frac{l}{\sqrt{a^2+ b^2}}\right).
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  3. #3
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    3D Trigonometry

    Hello bollsakkie

    Welcome to Math Help Forum!
    Quote Originally Posted by bollsakkie View Post
    Hi All!

    I have this nagging problem that I've been at it for a long time. I have attached a scanned sheet with the question. I really hope someone can help me out or atleast push me in the right direction.

    Thanks a bunch in advance!

    Akshay
    I'm still thinking about the rest of the question, but here's a start.

    The attached diagram shows the vertical plane containing the line OB. It is in this plane that the z-axis - and therefore also the line OB - is rotated through an angle \gamma about O. Note that E, the new position of the point B, is not vertically above B, but above a point D lying on OB. ED = l.

    Now OB = OE = \sqrt{a^2+c^2}

    So \gamma = \arcsin\left(\frac{l}{\sqrt{a^2+c^2}}\right)

    As far as the unit vector along Ow is concerned, its z-component is clearly \cos\gamma = \sqrt{\frac{a^2 +c^2 - l^2}{a^2+c^2}}.

    For the x- and y-components, consider the unit vector along OB: it is \frac{a}{\sqrt{a^2+c^2}}\vec{i} + \frac{c}{\sqrt{a^2+c^2}}\vec{j}. Then multiply this by -\sin\gamma\, (= - \frac{l}{\sqrt{a^2+c^2}}) for the component of the unit vector of Ow along the line OB.

    So I reckon the unit vector along Ow is -\frac{la}{a^2+c^2}\vec{i} - \frac{lc}{a^2+c^2}\vec{j}+ \sqrt{\frac{a^2 +c^2 - l^2}{a^2+c^2}}\vec{k}

    Grandad
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