I'll look at * vectors* from an
*algebraic* point of view and a *geometric* point of
view.

*Algebraically*, a * vector* is an ordered
list of (usually) real numbers. Here are some 2-dimensional vectors:

The numbers which make up the vector are the vector's * components*.

Here are some 3-dimensional vectors:

Since we usually use x, y, and z as the coordinate variables in 3 dimensions, a vector's components are sometimes referred to as its x, y, and z-components. For instance, the vector has x-component 1, it has y-component 2, and it has z-component -17.

The set of 2-dimensional real-number vectors is denoted , just like the set of ordered pairs of real numbers. Likewise, the set of 3-dimensional real-number vectors is denoted .

*Geometrically*, a vector is represented by an arrow. Here are
some 2-dimensional vectors:

A vector is commonly denoted by putting an arrow above its symbol, as in the picture above.

Here are some 3-dimensional vectors:

The relationship between the algebraic and geometric descriptions comes from the following fact: The vector from a point to a point is given by .

In 3 dimensions, the vector from a point to a point is .

* Remark.* You've probably already noticed the
following harmless confusion: " " can denote
the * point* in the x-y-plane, or the
2-dimensional real vector . Notice that the vector from the
origin to the point is the vector
.

So we can usually regard them as interchangeable. When there's a need to make a distinction, I will call it out.

* Example.* (a) Find the vector from to .

(b) Find the vectors , , , and for the points , , , and .

Sketch the vectors and .

(a)

(b)

Notice that ; this is true in general.

Here's a sketch of the vectors and :

and are both ; in the picture, you can see that the arrows which represent the vectors have the same length and the same direction.

*Geometrically*, two vectors (thought of as arrows) are * equal* if they have the same length and point in
the same direction.

* Example.* In the picture below, assume the two
lines are parallel. Which of the vectors , , is equal to the vector ?

is not equal to ; it has the same direction, but not the same length.

is not equal to ; it has the same length, but the opposite direction.

is equal to , since it has the same length and direction.

*Algebraically*, two vectors are * equal*
if their corresponding components are equal.

* Example.* Find a and b such that

Set the corresponding components equal and solve for a and b:

Substituting this into , I get , so .

The solution is , .

The * length* of a geometric vector is the length
of the arrow that represents it.

The * length* of an algebraic vector is given by
the distance formula. If , the length of
is

A vector with length 1 is called a * unit
vector*.

* Example.* (a) Find the length of .

(b) Show that is a unit vector.

(a)

(b)

*Algebraically*, you * add* or * subtract* vectors by adding or subtracting
corresponding components:

(Use an analogous procedure to add or subtract 3-dimensional vectors.) You can't add or subtract vectors with different numbers of components. For example, you can't add a 2 dimensional vector to a 3 dimensional vector.

*Algebraically*, you * multiply a vector by a
number* by multiplying each component by the number:

Vectors that are multiples are said to be *
parallel*.

* Example.* Compute:

(a) .

(b) .

(c) .

(d) .

(a)

(b)

(c)

(d)

Here are some properties of vector arithmetic. There is nothing surprising here.

* Proposition.* Let , , and be vectors (in the same space) and
let k be a real number.

(a) (Associativity) .

(b) (Commutativity) .

(c) (Zero vector) The vector with all-0 components satisfies and .

(d) (Additive inverse) The additive inverse of is the vector whose components are the negatives of the components of . It satisfies .

(e) (Distributivity) .

Note: To say that the vectors are *in the same space* means
that, for example, , , and are all vectors in
. But all of the results are true if , , and are vectors in
(100-dimensional Euclidean space).

* Proof.* The idea in all these cases is to write
the vectors in component form and do the computation. For example,
here is a proof of (c) in the case that .

Here is a proof of (e). I'll consider the special case where and are vectors in . Thus,

Then

The other parts are proved in similar fashion.

There is an alternate notation for vectors that is often used in
physics and engineering. , , and
are the * unit vectors* in the x, y, and z
directions:

Note that

For example,

In 2 dimensions, . There is no notation for vectors with more than 3 components.

You operate with vectors using the notation in the obvious ways. For example,

*Geometrically*, * multiplying a vector by a
number* multiplies the length of the arrow by the number. In
addition, if the number is negative, the arrow's direction is
reversed:

You * add* geometric vectors as shown below. Move
one of the vectors --- say --- keeping its length and
direction unchanged so that it starts at the end of the other vector.
Since the copy has the same length and direction as the original , it's equal to .

Next, draw the vector which starts at the starting point of and ends at the tip of . This vector is the sum .

The picture below illustrates why the geometric addition rule follows from the algebraic addition rule. It is obviously a special case with two 2-dimensional vectors with positive components, but I think it makes the result plausible.

To add several vectors, move the vectors (keeping their lengths and directions unchanged) so that they are "head-to-tail". In the second picture below, I moved and .

Finally, draw a vector from the start of the first vector to the end of the last vector. That vector is the sum --- in this case, .

The picture below shows how to subtract one vector from another --- in this case, is the vector which goes from the tip of to the tip of .

There are a couple of ways to see this. First, if you interpret this as an addition picture using the "head-to-tail" rule, it says

Alternatively, construct by "flipping" around, then add to .

This gives . As the picture shows, it is the same as the vector from the head of to the head of , because the two vectors are opposite sides of a parallelogram.

* Example.* Vectors and are shown in the picture below.

Draw pictures of the vectors , , and .

Copyright 2018 by Bruce Ikenaga