40. Introduction to vectors

Definition 40.1. A vector is a column of two, three, or more numbers, written

in general.
The length of a vector is defined by

We will always deal with either the two or three dimensional cases, in other words, the cases n = 2
or n = 3, respectively. For these cases there is a geometric description of vectors which is very useful. In
fact, the two and three dimensional theories have their origins in mechanics and geometry. In higher
dimensions the geometric description fails, simply because we cannot visualize a four dimensional
space, let alone a higher dimensional space. Instead of a geometric description of vectors there is
an abstract theory called Linear Algebra which deals with “vector spaces” of any dimension (even
infinite!). This theory of vectors in higher dimensional spaces is very useful in science, engineering
and economics. You can learn about it in courses like MATH 320 or 340/341.

40.1. Basic arithmetic of vectors

You can add and subtract vectors, and you can multiply them with arbitrary real numbers. this
section tells you how.

The sum of two vectors is defined by



The zero vector is defined by

It has the property that

no matter what the vector is.

You can multiply a vector with a real number t according to the rule

In particular, “minus a vector” is defined by

The difference of two vectors is defined by

So, to subtract two vectors you subtract their components,

40.2 Some GOOD examples

40.3 Two very, very BAD examples. Vectors must have the same size to be added, therefore

Vectors and numbers are different things , so an equation like

is nonsense!

This equation says that some vector () is equal to some number (in this case: 3). Vectors and numbers
are never equal!

40.2. Algebraic properties of vector addition and multiplication

Addition of vectors and multiplication of numbers and vectors were defined in such a way that the
following always hold for any vectors (of the same size) and any real numbers s, t

[vector addition  is commutative]
[vector addition is associative]
[first distributive property]
[second distributive property]

40.4 Prove (49). Let and be two vectors, and consider both possible ways of
adding them:

We know (or we have assumed long ago) that addition of real numbers is commutative, so that
, etc. Therefore

This proves (49).

40.5 Example. If and are two vectors, we define

Problem: Compute and in terms of and .


Problem: Find s, t so that

Solution: Simplifying you find

One way to ensure that holds is therefore to choose s and t to be the solutions of

The second equation says t = −3s. The first equation then leads to 2s + 3s = 1, i.e. . Since
t = −3s we get . The solution we have found is therefore

40.3. Geometric description of vectors

Vectors originally appeared in mechanics, where they represented forces: a force acting on some
object has a magnitude and a direction. Thus a force can be thought of as an arrow, where the length of
the arrow indicates how strong the force is (how hard it pushes or pulls).

So we will think of vectors as arrows: if you specify two points P and Q, then the arrow pointing
from P to Q is a vector and we denote this vector by .

The precise mathematical definition is as follows:

two pictures of
the vector
Definition 40.6. For any pair of points P and Q whose coordinates are and one defines
a vector by

If the initial point of an arrow is the origin O, and the final point is any point Q, then the vector is called the
position vector of the point

If and are the position vectors of P and Q, then one can write as

For plane vectors we define similarly, namely, . The old formula for the distance
between two points P and Q in the plane

says that the length of the vector is just the distance between the points P and Q, i.e.

distance from P to

This formula is also valid if P and Q are points in space.

40.7 Example. The point P has coordinates (2, 3); the point Q has coordinates (8, 6). The vector
is therefore

This vector is the position vector of the point R whose coordinates are (6, 3). Thus

position vectors in the plane
and in space


The distance from P to Q is the length of the vector , i.e.

distance P to


40.8 Example. Find the distance between the points A and B whose position vectors are
and respectively.

Solution: One has

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