3-dimensional coordinate systems [9.1]

Locating objects in 3-dimensions.

• Right-handed Cartesian coordinates
• Points in 3-D space
• Equations of planes
• Distance formula in 3 dimensions
• Equations of spheres

Rectangular coordinates

Also known as Cartesian coordinates, after Renes Descartes

Right-handed coordinate system

1. Take your right hand...
2. Point your straight fingers in the $\hat x$ direction.
3. Rotate your hand so that you can curl your fingers* to point in the $\hat y$ direction.
4. Your thumb now points in the $\hat z$ direction.

* You need to be able to curl your fingers through an angle less than $180^o$.

Try x and y, then x and z and y and z, and figure out which way the other axis points.

To Do

• Create a set of coordinate axes [pipecleaners]
• Do the Visualizing the xyz coordinate system worksheet (See the "handouts" folder):
  Write down which color is your $x$-axis, your $y$-axis, and your $z$-axis.

Coordinate planes

Coordinates

Projections

$S$ is the projection of $P$ onto the $xz$ plane.

$(0,b,0)$ is the projection of $P$ onto the $y$-axis.

The space $\mathbb{R}^3$

$\mathbb{R}\equiv$ the real numbers.

• Cartesian Product:
  $\mathbb{R}^3\equiv \mathbb{R}\times\mathbb{R}\times\mathbb{R}=\{(x,y,z) | x,y,z\in \mathbb{R}\}$.
• Set of all ordered triples.

To Do

• Points in the xyz coordinate system worksheet

Surfaces in $\mathbb{R}^3$

All points with coordinates $(a,b,c)$ where $c=3$
[and $a$ and $b$ can be anything]

All points with coordinates $(x,y,z)$ where $y=x$.

Notice that $y=x$ is the definition of a line in 2-d with a slope of 1. But here??

Anything a bit deceptive about the picture above?

Should continue to values with x and y less than 0...

To Do

• Section 9.1 Group Work: Fun with Visualization [missing]
• Working with surfaces in 3D space [missing]

Distance

What is the distance $|P_1P_2|$ between two points $P_1(x_1,y_1,z_1)$ and $P_2(x_2,y_2,z_2)$ in three dimensional space?

With $\Delta x=x_2-x_1$, etc, you can imagine applying Pythagoras' theorem twice [at GeoGebra, 9.1.distance.nb]:

• The blue line is the hypotenuse of the triangle with legs $\Delta x$ and $\Delta y$: It's length is $\sqrt{(\Delta x)^2+(\Delta y)^2}$.
• The green line is our distance, and is the hypotenuse of the right triangle with legs of $\Delta z$ and the blue line.

So, our distance formula is: $$|P_1P_2|=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}.$$

Equation of a sphere

All the points on a sphere are the same distance, $r$, away from its center, $C(h,k,l)$. Using the distance formula: $$(x-h)^2+(y-k)^2+(z-l)^2 = r^2.$$