Three-Dimensional Rectangular Geometry

Exercises

1. Plot the points \((1,4,-6)\) and \((-2,3,6)\) and \((3,-4,-5)\) in \(xyz\)-space and determine which one is closest to the origin.
2. What’s the distance between the points \((-6,7,-8)\) and \((0,21,-28)?\) What are the coordinates of the midpoint between those points?
3. Sketch the planes \(y=5\) and \(z=-3\) and \(x=y\) in \(xyz\)-space.
4. What’s an equation of the sphere with center \((4,5,-6)\) and radius three?

Problems & Challenges

1. What is the area of the triangle in space with vertices located at the coordinates \((1,1,1)\) and \((3,-3,8)\) and \((2,7,0)?\)
2. Consider the points \((1,1,1)\) and \((3,-3,8).\) Determine a point on the \(z\)-axis that is equidistant to those points.
3. Consider a line segment with one endpoint at the coordinates \((4,1,2)\) and midpoint at the coordinates \((-3,4,5).\) What are the coordinates of the other endpoint?