10.1.3 Formulas in Three Dimensions (continued)
Example 3 Graphing Figures in Three Dimensions
A cube is graphed in the three-dimensional coordinate space in this example. It is given that the cube has sides of 3 units and a vertex at the origin (0, 0, 0). The three dimensional coordinate space has three perpendicular axes called x , y , and z . By convention, the z -axis is vertical. To draw the cube, count three units along each of the axes to locate three of the vertices. Then from each of these vertices count three units along each of the other axes. This will locate the other vertices of the cube. Draw lines between adjacent vertices to represent the edges of the cube. A cylinder is graphed in the three-dimensional coordinate space in this example. It is given that the cylinder has radius of 2 units, height of 4 units, and one base centered at the origin (0, 0, 0). To draw the cylinder, count two units along the x - and y -axes in each direction away from the origin. Sketch the base of the cylinder as a circle with its circumference passing through the four intersection points with the x - and y -axes. Count up four units from the x - and y -axis intersection points to the upper base of the cylinder and sketch in the base centered on the z -axis. Draw the side of the cylinder by connecting the bases.
Example 4 Finding Distances and Midpoints in Three Dimensions
The Distance Formula in Three Dimensions is an extension of the Distance Formula in Two Dimensions. There is a term added for the third dimension representing the location of the two points on the z -axis. Notice that the formula is a form of the Pythagorean Theorem. The Midpoint Formula in Three Dimensions is an extension of the Midpoint Formula for Two Dimensions. The formula includes a term for the average of the location of the points on the z -axis.
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