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Math Help - Vector dot product question

  1. #1
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    Vector dot product question

    Hi, I've been given the following question for an assignment that carries marks, I've made an attempt and I got some answers, but they don't seem right. Can someone please help me. Thanks!!

    (a) A space diagonal of a cube is a line segment joining two diagonally opposite vertices of the
    cube in R^3. Find the angle between two space diagonals of a unit cube. - I got 70.53 degrees.


    (b) A few years ago it was feared that the Leaning Tower of Pisa was about to collapse. Suppose
    two steel cables are attached to the highest point P of the tower (which is 56:5m above ground
    level) and anchored to the ground at points 80m East and 20m North of P, and 40m East and
    50m North of P respectively, at the same level as the tower's base. What angle do the two cables
    make at P? - I got 30 degrees.
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  2. #2
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    Re: Vector dot product question

    (a) Let \vec{a} be the space diagonal and \vec{b} be the projection of that diagonal onto the x-y plane (so this is just the diagonal of the square on the bottom). Now, \vec{a}\cdot\vec{b} = \langle 1,1,1 \rangle \cdot \langle 1, 1, 0 \rangle = 2 . By the def of dot product this is |a||b|cos\alpha = \sqrt{3}\sqrt{2}\cos\alpha = \sqrt{6}\cos\alpha. I'll let you fill in the details.

    (b) Imagine your coordinate system with respect to the tower. What are the vectors formed by both wires? Please respond with your answer and we'll go from there.
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  3. #3
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    Re: Vector dot product question

    Thanks for your help. I got OA = 80i + 20 j - 56.5k and OB = 40i + 50j - 56.5k with the point P and origin.
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  4. #4
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    Re: Vector dot product question

    Quote Originally Posted by Vishak View Post
    Thanks for your help. I got OA = 80i + 20 j - 56.5k and OB = 40i + 50j - 56.5k with the point P and origin.
    That sounds correct. Now take the dot product of those two vectors, and this should equal to |\vec{OA}||\vec{OB}|cos\theta. Solve for \theta.
    Thanks from Vishak
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  5. #5
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    Re: Vector dot product question

    Hello, Vishak!

    I agree with both of your answers.

    Thanks from Vishak
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  6. #6
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    Re: Vector dot product question

    Okay, thanks a lot guys!!!!
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