Section 1.1 3-dimensional space
This is an introduction section for the three dimensional space. We recall some basics of two dimensional space first by working on some problems.
Exercise 1: Draw the graphs of the given equations.
(a) [latex]x+y=1[/latex]
(b) [latex]y=2[/latex]
(c) [latex]x=-3[/latex]
(d) The line(s) that is(are) parallel to the [latex]y[/latex]-axis and is(are) two units away from the [latex]y[/latex]-axis.
(e) The circle center at [latex](-1,2)[/latex] radius [latex]3[/latex]. 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 [latex]x[/latex]-axis, the [latex]y[/latex]-axis, and the [latex]z[/latex]-axis. Because each axis is a number line representing all real numbers in [latex]\mathbb{R}[/latex], the three-dimensional system is often denoted by [latex]\mathbb{R}^{3}[/latex]. We extend the [latex]xy[/latex]-coordinate system to the [latex]xyz[/latex]-coordinate system using right hand rule.
Example 2: Locate the points on the 3-dimensional coordinate system.
[latex]A=(0,2,1)[/latex], [latex]B=(-2,0,3)[/latex]. Find the length of line segments. [latex]\overline{AB}[/latex].
Exercise 3: Locate the points on the 3-dimensional coordinate system.
[latex]A=(-1,1,-3)[/latex] [latex]B=(0,1,3)[/latex], [latex]C=(2,-1,0)[/latex], [latex]D=(2,1,3)[/latex], [latex]E=(1,1,-3)[/latex] . Find the length of line segments. [latex]\overline{AB}[/latex], [latex]\overline{CD}[/latex], [latex]\overline{DE}[/latex].
Example 3: Draw the graphs of the given equations.
(a) The plane [latex]x=-3[/latex].(b) The plane [latex]y=2[/latex].
(c) The plane(s) that is(are) parallel to the [latex]xy[/latex]-plane and is two units away from the [latex]xy[/latex]-plane.
Exercise 3: Draw the graphs of the given equations.
(a) The plane [latex]x=-3[/latex].(b) The plane [latex]y=2[/latex].
(c) The plane(s) that is(are) parallel to the [latex]xy[/latex]-plane and is two units away from the [latex]xy[/latex]-plane.
Group Work: Draw the graphs of the given equations in 3D Space(a) The sphere center at [latex](0,0,0)[/latex] radius [latex]2[/latex]. Write down the sphere equation.
(b) The sphere center at [latex](-1,2,3)[/latex] radius [latex]3[/latex]. Write down the sphere equation.
(c) The sphere has [latex](1,2,3)[/latex] and [latex](3,0,5)[/latex] as end points of a diameter. Write down the sphere equation.
(d) What is the sphere equation with center [latex](a,b,c)[/latex] and radius [latex]r[/latex]?