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40. Introduction to vectors
|Definition 40.1. A vector is a column of
two, three, or more numbers, written
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 vectorwith 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
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
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
Definition 40.6. For any pair of points P and Q whose coordinates are
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
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
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.
The point P has coordinates (2, 3); the point Q has coordinates (8, 6).
This vector is the position vector of the point R whose coordinates are (6, 3). Thus
position vectors in the plane
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
Solution: One has