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
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 $\displaystyle (0,\sin\theta,\cos\theta)$.
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 $\displaystyle \gamma = {\textstyle\frac12}(\alpha+\beta)$ with the y-axis, and it's easy to see from this that the line AB will be in the direction $\displaystyle (\cos\gamma,\sin\gamma,0)$. Now all you have to do is to take the inner product of this vector with the one in the vertical direction.
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 ?
That's fine. Now notice that the line OM makes an angle $\displaystyle \textstyle\frac12(\alpha+\beta)$ with the axis Oy. Therefore AB makes the same angle with the axis Ox.
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.
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.