Problem: If I walk 5 miles to the East, then 8 miles to the North, I will have walked 13 miles. If you walk in a straight line from my starting point to my final position, what angle will your path make with East, and how far will you walk? At what angle, measured from East, should you walk?
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Solution: A sketch should consist of a right triangle with one leg to the North and the other to the East. The legs of the triangle represent 5 and 8 miles, and the hypotenuse represents the actual displacement. The actual distance is the length of the hypotenuse, which is found by the Pythagorean Theorem to be `sqrt( ( 5) ^ 2 + ( 8) ^ 2) miles = 9.433 miles.
The angle at which you should walk will be arcTan( 8 / 5) = 1.012 degrees.
Generalized Response: We can set up an x-y coordinate system with the x axis point East and the y axis pointing North. If we move through a displacement sx in the x direction then through a displacement sy in the y direction, we will end up at a point that corresponds to a vertex of the triangle formed by applying sx at the initial point, then sy at the terminal point of sx. The point at the other end of the hypotenuse from the starting point, and the displacement will be along the hypotenuse, which has length `sqrt(sx^2 + sy^2) and lies at angle arctag(sy / sx) from the origin.
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Figure description: The figure below shows the displacements sx and sy as vectors parallel to the x and y axes. The path that results from starting at the origin and moving through displacement sx followed by displacement sy is indicated by the heavy (blue) lines forming the legs of the triangle. The displacement is the vector dS from the origin (the starting point of the path) to the terminal point of the path.
The effective distance of this displacement is the length `sqrt(sx^2 + xy^2) of the hypotenuse, and the angle of the displacement is arctan(sy / sx).
