1.2 Three Dimensional Space

Three Dimensional Space

Introduction

Three Dimensional Space and Coordinates

Fundamental Planes

Distance in Three Dimensions

In this Preview Activity, we will recall some ideas about measurements and important ideas in two dimensional space.

What are the coordinates of the point drawn in ? Draw segments from the point $P$ to the vertical and horizontal axes to demonstrate the measurements of the coordinates.

Find the coordinates of the point $A$ where one ends if one starts at point $(1,2,3)$ and moves 5 units forward, 4 units to the left, and 2 units up.

Draw the point $A$ (that is, your answer to ()) on a set of three-dimensional axes and include the line segments that show that coordinate's points as a set of directions from the origin (like in Figure 1.2.7 with the “Show Segments” option).

Find the coordinates of the point $B$ where one ends if one starts at point $(3,-4,2)$ and moves 4 units backwards, 4 units to the right, and 4 units down.

Draw the point $B$ (that is, your answer to ()) on a set of three-dimensional axes and include the line segments that show that coordinate's points as a set of directions from the origin (like in Figure 1.2.7 with the “Show Segments” option).

Consider the set of points $(x,y,z)$ that satisfy the equation $x=2$. Write a sentence to describe this set as best as you can.

Consider the set of points $(x,y,z)$ that satisfy the equation $y=-1$. Write a sentence to describe this set as best as you can.

Consider the set of points $(x,y,z)$ that satisfy the equation $z=0$. Write a sentence to describe this set as best as you can.

Consider the right triangle $PRS$ in the base of the box whose hypotenuse is shown as the red line in . What are the coordinates of $R$ and $S$?

Give the equation of the fundamental plane that contains the right triangle $PRS$.

Since the right triangle $PRS$ lies in a plane, we can use the Pythagorean Theorem to find a formula for the length of the hypotenuse of this triangle. Find the length of the segment $PR$ in terms of $x_0$, $y_0$, $x_1$, and $y_1$.

Triangle $PRQ$ has hypotenuse drawn with as blue segment connecting the points $P$ and $Q$. Segment $PR$, which is the hypotenuse of triangle $PRS$ that we considered earlier, is a leg of triangle $PRQ$. This triangle lies entirely in a plane, so we can again use the Pythagorean Theorem to find the length of its hypotenuse. Show that the length of $PQ$ is

In this exercise, we want to graphically identify when a given plot can be expressed as: -$y$ as a function of $x$, but $x$ cannot be written as a function of $y$-$x$ as a function of $y$, but $y$ cannot be written as a function of $x$-$y$ as a function of $x$ and $x$ as a function of $y$- neither $y$ as a function of $x$ nor $x$ as a function of $y$

For each graph in , state which of the four options above applies to whether the graph can be expressed with each coordinate being a function of the other or not. If the graph cannot be expressed with $y$ as a function of $x$, write a sentence about what property you used to determine this. If the graph cannot be expressed with $x$ as a function of $y$, write a sentence about what property you used to determine this.