Section 1.1 3-dimensional space

This is an introduction section for the three dimensional space. We recall some basics of two dimensional space  by working on some problems.

Exercise 1: Draw the graphs of the given equations.

(a) $x+y=1$

(b) $y=2$

(c) $x=-3$

(d) The line(s) that is(are) parallel to the $y$-axis and is(are) two units away from the $y$-axis.

(e) The circle center at $(-1,2)$ radius $3$. Write down the circle equation.

(f) The parabola

$$y=x^2-2$$

(g) The parabola

$$y^2=x-2$$

Example 1: Draw the graphs of the given equations.

(a)The ellipse 

$$\frac{(x-2{)}^2}{2^2}+\frac{(y+1{)}^2}{3^2}=1$$

(b) The hyperbola 

$$\frac{(x-2{)}^2}{2^2}-\frac{(y+1{)}^2}{3^2}=1$$

(c) The hyperbola 

$$-\frac{(x-2{)}^2}{2^2}+\frac{(y+1{)}^2}{3^2}=1$$

Three-Dimensional Coordinate Systems

Definition: The three-dimensional rectangular coordinate system consists of three perpendicular axes: the $x$-axis, the $y$-axis, and the $z$-axis. Because each axis is a number line representing all real numbers in $\mathbb{R}$, the three-dimensional system is often denoted by ${\mathbb{R}}^3$. We extend the $xy$-coordinate system to the $xyz$-coordinate system using right hand rule.

Example 2: Locate the points on the 3-dimensional coordinate system.

$A=(0,2,1)$, $B=(-2,0,3)$. Find the length of line segments. $\overline{AB}$.

Exercise 3: Locate the points on the 3-dimensional coordinate system.

$A=(-1,1,-3)$ $B=(0,1,3)$, $C=(2,-1,0)$, $D=(2,1,3)$, $E=(1,1,-3)$ . Find the length of line segments. $\overline{AB}$, $\overline{CD}$, $\overline{DE}$.

Example 3: Draw the graphs of the given equations.

(a) The plane $x=-3$.

(b) The plane $y=2$.

(c) The plane(s) that is(are) parallel to the $xy$-plane and is two units away from the $xy$-plane.

Exercise 3: Draw the graphs of the given equations.

(a) The plane $x=-3$.

(b) The plane $y=2$.

(c) The plane(s) that is(are) parallel to the $xy$-plane and is two units away from the $xy$-plane.

Group Work: Draw the graphs of the given equations in 3D Space

(a) The sphere center at $(0,0,0)$ radius $2$. Write down the sphere equation.

(b) The sphere center at $(-1,2,3)$ radius $3$. Write down the sphere equation.

(c) The sphere has $(1,2,3)$ and $(3,0,5)$ as end points of a diameter. Write down the sphere equation.

(d) What is the sphere equation with center $(a,b,c)$ and radius $r$?

(e) Draw the plane $x+y=3$