Is there a name to refers to anything that is a point, line, plane, etc?

Question

I'm teaching my juniors in high school some beginning linear algebra, but I find there is some vocabulary I am missing. I want to say that points, lines, and planes are all related, but is there a name for them? I keep wanting to say "linear object," but I have a feeling that that means something else.

Answer 1

Your Question Was:

Is there a name to refers to anything that is a point, line, plane, etc?

Answer:

Yes, there is a name for such things.

All of the following subsets of Cartesian space are examples of Vector Spaces

• a point (all by itself)
• a line of points.
• a plane.

What is a Vector Space?

The following is optional reading if you only wanted to know what the name was.

Informally, vector space is a set of points such that:

1. The space a center-point (usually called the "zero vector")
2. The space has has no holes or missing pieces.
3. The space has no ends (it stretches away in some directions forever and ever).

When dealing with a vector spaces, mathematicians talk about vectors more than the points inside of the space. For each point floating around in space, the vector associated with that point is a simple instruction which specifies how to travel that point from the origin. Usually, a vector is an ordered pair of two numbers, such as $(4.12, 6.90)$, but it varies from space to space. A vector says, "add this amount to the zero vector, and you will arrive at the point I represent."

That is, when working with vector spaces, each point in space is encoded as a way to travel to that point from the origin.

Multiplication and Addition

Vector spaces have two special definitions of:

• How to "add" two vectors together
• How to "multiply" a vector by a decimal number.

Vector-multiplication represents how to grow or shrink a thing.

Vector-addition represents how to translate (or move) a thing from one place to another.

Multiplication

Enlarging a photocopy is the best analogy I have for understanding what happens when you multiply a vector by a decimal number.

In order to view a line as a vector-space, you have to pick a single dot on the line to be the center of the line (the "zero vector"). Multiplying any dot on the line by a decimal number amounts to sliding the dot along the line until the dot is further away, or closer to, the center of the line than it used to be.

Multiplying a vector by a number is almost the same as increasing or decreasing the radius-part of a coordinate written in polar-coordinates.

If you multiply a vector $v$ in a two-dimensional vector-space by $0.43$, then you just moved the point associated with vector $v$ so that it ended up closer to the origin than where it started.

To grow a shape inside of a vector space you multiply all of the vector in the shape multiply by a decimal number greater than 1.

End Description of Vector Multiplication

Begin Description of Vector Addition

If someone asks you to add two vectors together with a plus sign (+) then that is equivalent to giving you:

1. a point to be moved
2. a direction to move the point
3. a distance to move the point.

For two vectors $v$ and $w$, $v + w$ is the point in space you arrive at after you:

1. begin at the origin (zero vector).
2. travel in a straight line some distance and direction specified by vector $v$.
3. stop
4. travel in a straight line some distance and direction specified by vector $w$

That is, a vector says that a point is reachable from the origin by describing a one-step journey.

The addition of two vectors describes a point reachable in a two-step journey.

Vector-multiplication is how we grow and shrink a cloud of dots centered at the origin.

Vector-addition is how we translate (slide) a cloud of dots from one place to another without shrinking or growing the cloud of dots.

End Description of Vector Addition

It Never Ends (Infinite Multiplication)

If you begin with a rectangle and stretch it out flat until it is infinite in surface area, then the result is a vector space.

If you begin with a sphere and then expand the sphere in all directions, until the sphere is infinite in size, then the result is a vector space.

A property of vector multiplication is that for any distance (no matter how large) if you travel that distance away from the origin, then you are still inside of that vector-space.

If you travel $200$ miles away from the zero-vector, then you are still inside of the space.

If you travel $9999999999$ million miles away from the zero-vector (aka "center") of the space, then you will still be inside of the same space.

Vector-spaces stretch on forever and ever without ending because there is no limit to the size of the multiplication (enlargement) you can perform.

Instead of saying that a vector space is infinitely large, mathematicians instead created an imaginary robot which always says "yes" if you ask the robot if everything $x$ miles away from the origin is still inside of the space.

• Are points 5 miles away from the center still inside of the space? Yes.
• Are points 50 miles away from the center still inside of the space? Yes.
• Are points 500 miles away from the center still inside of the space? Yes.

The answer is always "yes," no matter how far you go.

There Are No Holes (Infinite Addition)

Suppose that you drew a very large letter "X" on a piece of paper. Vector-multiplication will move the tips of the letter "X" away from the center until the "X" is infinite in size. The resulting set of point would probably not be viewed as a vector-space. This is because after you leave the origin (zero-vector) the only direction you are allowed to travel is in a straight arrow pointing away from the origin, or you can walk back toward the origin.

Vector addition allows us to move in any direction we like.

Suppose that $XS$ is a "vector - space." All vector spaces have the same property as $XS$, where the property is...

If we do the following pick two random "vectors" $x$ and $y$ in space $XS$, and add $x + y$ together, then the result will be a point which is also inside of the same space.

For example, suppose that we began with a dozen dots evenly spaced-out on a circle centered at the origin of a plane. Vector-Multiplication turns the dots into things which look like lines or sun-rays.

Vector-addition lets us rotate the points. The rotation is a little awkward, because we will travel in a straight line instead of a curving arc along the edge of a circle. However, it is possible to convert rotation from degrees/radians into something you can add to an x-y coordinate pair in order to move a dot from one place on the circle, to another.

Vector addition can be used to implement translations and rotations.

Vector multiplication represents scaling a shape up or down.

Two Examples of Really Weird Vector Spaces

Example One

An finite set of isolated dots on the plane can be a vector space.

• You can define a successor function $S$ such that for any dot $x$ points $S(x)$ is the next dot. The successor function can form a cycle and eventually loop back around.

• To multiply a vector by decimal number you can round the decimal number to the nearest whole number. After that, input the integer into the iterated successor function. For any vector $v$ and $\forall \alpha \in \mathbb{Z}, S^{\alpha}(v) = S(S^{\alpha - 1}(v))$. $S^{0}(v) = v$

• In order to add two vectors $v$ and $w$ you can find a whole number $\alpha$ such that $S^{\alpha}(v) = w$. Then, $v+w = \alpha*w$. If $v$ or $w$ are the zero vector then we let $\alpha$ be the decimal number zero.

End First Example of a Really Weird Vector Space

Begin Second Example of a Really Weird Vector Space

Anything can be vector space if you define vector-addition and vector-multiplication in the right way.

There are a few restrictions, but when we define a vector-space we are allowed to define vector-multiplication and vector-addition however we like.

The following examples show just how different vector-multiplication and vector-addition can be from the multiplication and addition we are used to.

A "graph" is a network of dots connected by lines. Specifically, there are finite number of dots connected by lines such that there is at most one line between every pair of dots. If you draw the dots on a sheet of paper, then the exact location (or position) of the dots on the plane does not matter; it is considered to be the same graph regardless of how you move the dots.

We could define a vector-space as follows:

• Let the set of vectors be the set of all graphs $G$ such that every node in graph $G$ has at most $2$ lines coming out of it. That is, a vector is the disjoint union of one or more paths and cycles. The vertex sets of these graphs shall always be sets of decimal numbers. We allow the vertex set to be the empty set.

• For any two graphs $G$ and $H$ we will write $G = H$ only if the vertex set of graph $G$ is the same as the vertex set of graph $H$. The edge sets should also be exactly the same, not merely isomorphic.

• Define addition $(+)$ between vectors such that $\forall G, H \in \mathcal{THE} \text{ } \mathcal{VECTORS}, \quad$ $G + H$ is equal to a special type of union of graphs $v$ and $w$. If a vertex is the same number in graph $G$ and graph $H$ then that vertex should be just one node, not two nodes, in the union of the two graphs. The vertex set of $G + H$ is the set-union of the vertex set of $G$ and the vertex set of graph $H$ except that for any decimal number $x$, if $x$ is a vertex in graph $G$ and $(-1)*x$ is a vertex in graph $H$ then $x$ not in the vertex set of $(G + H)$ and $(-1)*x$ not in the vertex set of $(G + H)$. The edge set of graph $(G + H)$ shall be $\{\{x, y\}: x,y \in (G+H).VS \text{ and } \{x, y\} \in G.ES \text{ or } \{x, y\} \in H.ES \}$

• For every vector $v$ in the space, we will define multiplication between numbers and vectors such that for any decimal number $\alpha$ and vector $G = (G.VS, G.ES)$, we will say that $\alpha * G$ is the graph $H = (H.VS, H.ES)$ such that $H.VS = \{\alpha * V: V \in G.VS\}$ and $H.ES = \{(\alpha * V_{1}, \alpha * V_{2}): V_{1}, V_{2} \in G.VS \text{ and } \{V_{1}, V_{2}\} \in G.ES\}$

• We have the associative property. $v + (w + x) = (v + w) + x$. The sums of graphs are equal to the unions of graphs.

• We have commutativity of vector addition: for any vectors (graphs) $v$ and $w$, $v + w = w + v$ because the union of two graphs is commutative.

• We have the identity element of vector addition. There exists an element $0 ∈ \mathcal{THE} \text{ } \mathcal{SPACE} \qquad$, called the zero vector, such that $v + 0 = v$ for all $v ∈ \mathcal{THE} \text{ } \mathcal{SPACE} \qquad$. Specifically, $0$ is the graph whose vertex set is the empty set.

• We have the inverse property of vector addition. $v + (−1)*v$ is the zero-vector.

Last Part of the Anything-Can-Be-a-Vector-Space

The Anything-Can-Be-a-Vector-Space section was overly confusing and complicated.

The point was supposed to be that anything in mathematics can be looked at as if it was a vector space.