# Cartesian Coordinate Exercise Solutions

1. #### Right handed axis system

(The positive y-axis points into the page.)

#### Left handed axis system

(The positive y-axis points into the page.)

2. The cartesian coordinates of the origin are (0, 0, 0). The cartesian coordinates of a point on the x axis would have the form (x, 0, 0) for some x. The coordinates of a point on the plane containing the y and z axes would look like (0,  yz).

4. #### The Three-Dimensional Pythagorean theorem

If a point P has coordinates (x1x2x3), then its distance from the origin is

Proof: Project the point P onto the x-y plane and call this projection P'. Project P' onto the x axis and call this projection P''. Now we have two right triangles to consider:

Let's consider the second of these first. Now, the leg of this triangle from the origin O to the point P'' has length |x1| and the leg from P' to P'' has length |x2|, so by the normal (two-dimensional) Pythagorean theorem, the hypotenuse of this triangle, which is the length of the segment from the orign to P', is

But the segment from P' to P has length |x3|, so by applying the Pythagorean theorem once again, we get that

as desired.

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