$\newenvironment {prompt}{}{} \newcommand {\ungraded }{} \newcommand {\todo }{} \newcommand {\oiint }{{\large \bigcirc }\kern -1.56em\iint } \newcommand {\mooculus }{\textsf {\textbf {MOOC}\textnormal {\textsf {ULUS}}}} \newcommand {\npnoround }{\nprounddigits {-1}} \newcommand {\npnoroundexp }{\nproundexpdigits {-1}} \newcommand {\npunitcommand }{\ensuremath {\mathrm {#1}}} \newcommand {\RR }{\mathbb R} \newcommand {\R }{\mathbb R} \newcommand {\N }{\mathbb N} \newcommand {\Z }{\mathbb Z} \newcommand {\sagemath }{\textsf {SageMath}} \newcommand {\d }{\mathop {}\!d} \newcommand {\l }{\ell } \newcommand {\ddx }{\frac {d}{\d x}} \newcommand {\zeroOverZero }{\ensuremath {\boldsymbol {\tfrac {0}{0}}}} \newcommand {\inftyOverInfty }{\ensuremath {\boldsymbol {\tfrac {\infty }{\infty }}}} \newcommand {\zeroOverInfty }{\ensuremath {\boldsymbol {\tfrac {0}{\infty }}}} \newcommand {\zeroTimesInfty }{\ensuremath {\small \boldsymbol {0\cdot \infty }}} \newcommand {\inftyMinusInfty }{\ensuremath {\small \boldsymbol {\infty -\infty }}} \newcommand {\oneToInfty }{\ensuremath {\boldsymbol {1^\infty }}} \newcommand {\zeroToZero }{\ensuremath {\boldsymbol {0^0}}} \newcommand {\inftyToZero }{\ensuremath {\boldsymbol {\infty ^0}}} \newcommand {\numOverZero }{\ensuremath {\boldsymbol {\tfrac {\#}{0}}}} \newcommand {\dfn }{\textbf } \newcommand {\unit }{\mathop {}\!\mathrm } \newcommand {\eval }{\bigg [ #1 \bigg ]} \newcommand {\seq }{\left ( #1 \right )} \newcommand {\epsilon }{\varepsilon } \newcommand {\phi }{\varphi } \newcommand {\iff }{\Leftrightarrow } \DeclareMathOperator {\arccot }{arccot} \DeclareMathOperator {\arcsec }{arcsec} \DeclareMathOperator {\arccsc }{arccsc} \DeclareMathOperator {\si }{Si} \DeclareMathOperator {\scal }{scal} \DeclareMathOperator {\sign }{sign} \newcommand {\arrowvec }{{\overset {\rightharpoonup }{#1}}} \newcommand {\vec }{{\overset {\boldsymbol {\rightharpoonup }}{\mathbf {#1}}}\hspace {0in}} \newcommand {\point }{\left (#1\right )} \newcommand {\pt }{\mathbf {#1}} \newcommand {\Lim }{\lim _{\point {#1} \to \point {#2}}} \DeclareMathOperator {\proj }{\mathbf {proj}} \newcommand {\veci }{{\boldsymbol {\hat {\imath }}}} \newcommand {\vecj }{{\boldsymbol {\hat {\jmath }}}} \newcommand {\veck }{{\boldsymbol {\hat {k}}}} \newcommand {\vecl }{\vec {\boldsymbol {\l }}} \newcommand {\uvec }{\mathbf {\hat {#1}}} \newcommand {\utan }{\mathbf {\hat {t}}} \newcommand {\unormal }{\mathbf {\hat {n}}} \newcommand {\ubinormal }{\mathbf {\hat {b}}} \newcommand {\dotp }{\bullet } \newcommand {\cross }{\boldsymbol \times } \newcommand {\grad }{\boldsymbol \nabla } \newcommand {\divergence }{\grad \dotp } \newcommand {\curl }{\grad \cross } \newcommand {\lto }{\mathop {\longrightarrow \,}\limits } \newcommand {\bar }{\overline } \newcommand {\surfaceColor }{violet} \newcommand {\surfaceColorTwo }{redyellow} \newcommand {\sliceColor }{greenyellow} \newcommand {\vector }{\left \langle #1\right \rangle } \newcommand {\sectionOutcomes }{} \newcommand {\HyperFirstAtBeginDocument }{\AtBeginDocument }$

We talk about basic geometry in higher dimensions.

The word geometry can be broken into geo meaning “world” and metry meaning “measure.” In this section we will tell you what our mathematical “world” is, and how we “measure” it.

### Higher dimensions

In our previous courses, we studied functions where the input was a single real number and the output was a single real number. Note, the word “real” is being used in a technical sense:

Which of the following are real numbers?
$\sqrt {7}$ $0$ $-3$ $\pi$ $e$ $x$ $\sqrt {-1}$ $\infty$

When we say a function maps a real number to a real number, we write: When working in two dimensions, we need a way of talking about ordered pairs of numbers. We denote the set of all ordered pairs of real numbers by $\R ^2$. When working in three dimensions we denote the set of all ordered triples of real numbers by $\R ^3$. In three dimensions we have three coordinates axes, the $x$-axis, $y$-axis, and $z$-axis:

The axes point according to the right-hand-rule: Of course you will need to “spin” your hand around to align your pointer-finger with the $x$-axis and your middle-finger with the $y$-axis. Then your thumb will point in the $z$-direction.
Which of the following axes are aligned according to the right-hand rule?
Point the “pointer finger” of your right hand in the positive direction of the $x$-axis while simultaneously pointing your “middle finger” in the positive direction of the $y$-axis. Your thumb will point in the positive direction of the $z$-axis.    ### Basic plotting

To plot a point $(a,b,c)$ in $\R ^3$, you move $a$ in the $x$-direction, $b$ in the $y$-direction and $c$ in the $z$ direction.

Of course, we’re going to be plotting many points. We typically described groups of points, as those that satisfy a given equation involving $x$, $y$, and $z$. Here is a place where working in three dimensions is really different from working in two.

• In $\R ^2$, any equation involving $x$ and/or $y$ draws a curve.
• In $\R ^3$, any equation involving $x$, $y$, and/or $z$ draws a surface.

The most basic surface in $\R ^3$ is a plane.

Which equation describes the $(y,z)$-plane?
$x=0$ $y=0$ $z=0$
For every point on the $(y,z)$-plane, the $x$-coordinate is zero.
Can you describes the solution set of $y=2$ in $\R ^3$?
It’s a horizontal line. It’s a vertical line. It’s a plane parallel to the $(x,y)$-plane. It’s a plane parallel to the $(x,z)$-plane. It’s a plane parallel to the $(y,z)$-plane.
$y=2$ consists of all those points where $y=2$, but $x$ and $z$ are allowed to be anything.

Another way to think of the point $(a,b,c)$ is as the intersection of the planes $x=a$, $y=b$, $z=c$.

Move the point around below to see the planes that define it.

While three planes need not intersect at all, the intersection of two (nonparallel) planes is a line.

What is the intersection of the $(x,y)$-plane and the $(y,z)$-plane?
The $x$-axis. The $y$-axis. The $z$-axis.
What is the intersection of the $(x,y)$-plane and the $(x,z)$-plane?
The $x$-axis. The $y$-axis. The $z$-axis.
What is the intersection of the $(x,z)$-plane and the $(y,z)$-plane?
The $x$-axis. The $y$-axis. The $z$-axis.

### Distance and spheres

So the objects in our geometry are made of points, and now we must tell you how we plan to “measure” objects. To do this, we’ll use our old friend, the distance formula.

What is the distance between the points $P=(-17,379)$ and $Q=(14,-101)$ in $\R ^2$?

On a completely related note, what’s the most famous theorem in mathematics? I’ll tell you: The Pythagorean Theorem. In essence, the distance formula is The Pythagorean Theorem. Let’s see if we can explain why the Pythagorean Theorem is true.

The distance formula also extends to higher dimensions:

What is the distance between the points $P=(0,0,0)$ and $Q=(1,1,1)$ in $\R ^3$?

In general we can extend this notion of distance to $\R ^n$:

What is the distance between the points $P=(0,0,0,0)$ and $Q=(1,1,1,1)$ in $\R ^4$?

#### Circles and spheres, disks and balls

Let me remind you what the definition of a circle is:

Is the center point of a circle part of the circle?
yes no

From the definition of a circle, we see that it is intimately related to the distance formula. Indeed, it is also the case the equation of a sphere is related to the distance formula in $\R ^3$:

The equation $(x-1)^2+y^2+(z+2)^2 = 4$ has a solution set in $\R ^3$ which forms a sphere. What is the center and radius of this sphere?
The expression $(x-1)^2+y^2+(z+2)^2$ is the square of the distance from $(1,0,-2)$ to $(x,y,z)$. If the square of the distance is $4$, then that distance is $2$. Since the solution set of this equation is all points which are a distance of $2$ away from $(1,0,-2)$, then this is a sphere of radius $2$ centered at $(1,0,-2)$.

Things really get interesting when we have both spheres and planes around. Spheres can intersect planes at no points (if they are missing the plane), at one point (if they are “just” touching the plane), or at an infinite number of points (here the intersection is a circle).

Let’s see some examples.

That wasn’t too bad, let’s see another.

For our last example, we’ve left the easiest case of all.