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Math Help - Walking Up a Hill

  1. #1
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    Walking Up a Hill

    I am good with the first part of this question. I just Cant seem to pull out the final result about the angle of the line joining the two people, This trig just got ugly for me.

    -Bobak
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  2. #2
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    Take coordinate axes as follows. The x- and y-axes are both in the inclined plane (that's the key thing), with the x-axis horizontal and the y-axis up the line of greatest slope. The z-axis will then be perpendicular to the plane, so it will not be vertical. In fact, a vertical line will be in the direction of the vector (0,\sin\theta,\cos\theta).

    If Arthur is at the point A and Bertha is at B, then the line joining O to the mid-point of AB will make an angle \gamma = {\textstyle\frac12}(\alpha+\beta) with the y-axis, and it's easy to see from this that the line AB will be in the direction (\cos\gamma,\sin\gamma,0). Now all you have to do is to take the inner product of this vector with the one in the vertical direction.
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  3. #3
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    Opalg I don't really understand why this problem needs to be done with 3 dimensions or vectors.

    I have attached my working for the first part. I am fairly sure the last result should be be pulled out form a bit of trig using the diagram i drew. Or am i barking up the wrong tree ?
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  4. #4
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    that looks right.
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  5. #5
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    That's fine. Now notice that the line OM makes an angle \textstyle\frac12(\alpha+\beta) with the axis Oy. Therefore AB makes the same angle with the axis Ox.

    Quote Originally Posted by bobak View Post
    Opalg I don't really understand why this problem needs to be done with 3 dimensions or vectors.
    An inclined plane is an intrinsically 3-dimensional concept, so I don't think you can avoid using 3 dimensions in the solution. The simplest way to find the angle that AB makes with the vertical is the method I suggested in my previous comment.
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  6. #6
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    I understand how the inner product of the two vectors give the results I am just having problems defining the vectors properly in the co-ordinate system. This is mainly cause I am struggling with defining the x y and z axis properly on a diagram. This may be asking for a bit much but could you possibly show me a little sketch of this.
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  7. #7
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    Quote Originally Posted by bobak View Post
    I understand how the inner product of the two vectors give the results I am just having problems defining the vectors properly in the co-ordinate system. This is mainly cause I am struggling with defining the x y and z axis properly on a diagram. This may be asking for a bit much but could you possibly show me a little sketch of this.
    Not sure if this helps. It's supposed to show why the vertical line is in the direction of the vector (0,\sin\theta,\cos\theta).
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  8. #8
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    That is very clear now. Thank you very much Opalg. I really do appreciate you help.
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