Introduction
As a course, Physics 2000, is quite likely to be distinctly different from your first
introductory course in physics, particularly if your first introductory course was
Physics 1000 or Physics 1050. This is neither good nor bad, it is simply
something to be aware of as the course begins, and not to loose sight of as the
course progresses.
The difference lies in your experience. In a course like Physics 1000 for example
your intuitive experience with things like motion or forces was probably helpful as
you made your way through the mathematical expressions for these quantities.
Your ability to mentally visualize various interactions based on your experience
while solving physics problems would have also been advantageous. In essence
you had some, if not a lot, of intuition to guide you as you worked to gasped
concepts, and conquer mathematical representations of physical interactions.
Your intuitions about electricity and magnetism are probably not as well
developed as those about motion etc. So what tools will you use to develop the
proper concepts? Ultimately, you will need to rely more heavily on mathematics
than was the case in your first introductory physics course.
A quick example: F = qv X B. You may, or may not, have seen this equation
before, none the less, it describes quite concisely the interaction between a
charged particle and a magnetic field (it is also the basis for constructing a
television). This equation tells you that as a charged particle enters a region of
space containing a magnetic field it will experience a force that moves the
particle in a direction that is perpendicular to the plane containing the v and B
vectors. For example, if the particle has a positive charge and is traveling across
this page from left to right, and the magnetic field is oriented upwards on the
page, then the force of the magnetic field on the particle will direct it out of the
page toward you.
If you are not familiar or comfortable with the vector cross product (X) then you
may have some difficulty understanding the details of the motion of charged
particles in magnetic fields. Compounding the situation, an understanding of the
vector cross product hinges on the use of unit vector notation. In other words,
understanding the physics is difficult enough, if your mathematical background is
incomplete things only get harder and harder.
Before you panic there are a couple of things to remember:
1. – You do not need to be an expert this instant.
2. – If you work consistently at filling in the weak spots you will be an expert, or
at least be able to survive.
The most important thing you can do right now is to be aware that mathematics
will be an important part of physics 2000, and to regularly schedule study time to
improve your skills.
We will take some time in the lab to review the mathematical essentials
beginning with vectors and continuing to calculus.
Vectors
A vector is a mathematical description of a quantity for which both the magnitude
and the direction of the quantity are important and must be taken in to account (by
contrast a scalar has only one important quantity, its magnitude). Since vectors are
different than scalars they require a distinct notation. Typically, letters are used to
denote variables and/or constants. For scalars a letter such as x, or y, or π is
normal
practice. To distinguish vectors, for example the “vector A”, representations

like, A , A , or A are commonly used.
Vectors usually find themselves in a coordinate system. For introductory physics
the coordinate system is almost always the Cartesian coordinate system or
reference frame. This is the familiar X, Y grid.
Cartesian Coordinate System
Two Dimensions
Y
A location
denoted by a
specific value
of (X,Y)
Note that the axes are perpendicular to each other and
Therefore completely distinct form one another. No value
of Y (except 0) intersects the X axis and vice versa. This
brings up a subtle but important point about coordinate frames.
If you pause for a moment and think about how you refer to
this coordinate frame you might find yourself referring to the
X
X and Y directions. This is a habit which while useful in the past
will need to change for Physics 2000. X and Y are not directions,
they never where. X and Y are locations. We will need something else to denote
direction when we work with vectors.
Three Dimensions
As you can see all the axes are mutually perpendicular. Y
To specify a location in three dimensions you will need
three coordinates (scalars) X,Y,Z.
X
Z
Unit Vectors
Now back to the issue of scalars and vectors. Given that direction and magnitude are
equally important and must be accounted for in vector operations we will require
a more precise and usable notation to work with that one that employs X and Y
as directional information. It is quite likely that you have used a notation in the
past like P = Px + Py (remember, bold face characters are one way of
representing vector quantities). This notation implies that X and Y are directions
which is not true.
If you think of the vector P as representing a move
from the origin of the reference frame to the point (X,Y)
then clearly the path taken by the vectors Px and Py is
equal to P in the sense that they end in the same
location, clearly P = Px + Py
Y
P
Py
Examine the vector Px. It travels some distance along
the X axis. That distance is represented by the value of
X in the order pair (X,Y) which is a scalar. Since X is not
a direction it makes some sense to use Px (note, not
bolded therefore not a vector) to represent that quantity
and to use something else to designate the direction.
Px
X
A vector can be considered to be the product of a scalar and another vector. For
example, if you take five steps in a particular direction that could be described as
taking “one step in that direction five times”. The one step is the vector, doing it
five times is multiplying the vector by the scalar number, five. The result is a
vector of magnitude five in some direction. This idea is the basis for unit vectors.
If we define a vector that has a magnitude of one in any direction we choose then
that vector contains only directional information. We can then multiply that vector
by any scalar (magnitude) we wish to produce a vector description of some
physical quantity. This notation for vectors has the important feature that it
completely separates magnitude information from directional information.
By convention the three axes of the Cartesian
coordinate frame (X,Y,Z) are given directions i, j, k.
Now the directions of the axes are distinct from locations
along the axes.
Y
j
X
k
i
Z
The vector P from the previous example can now be expressed as P = Px i + Py j.
Make careful note of what is a vector and what is a scalar in this equation.
Y
Px is a point on the X axis. The vector Px i is the displacement
from the origin to the point Px along the X axis,
which is the “i direction.”
P
Pyj
The vector P could also be expressed as Pp, where p is a unit
vector in the direction of P, and P is the magnitude of P.
The unit vector p has components
Like any other vector and can be expressed
as p = px i + py j. The values of px and py
contain special information in that they will
are equal to the value of the cosine of the angle
The value of px will equal cos x and py will
equal cos y.
Y
y P
p
x
Pxi
X
Any vector can be expressed in unit vector notation i.e. the vector A can be
expressed A = Ax i+ Ay j + Az k = Aa. The coefficients of i, j, k are the
magnitudes of the components of A along the respective axes of the coordinate
frame and can be used to calculate the magnitude of A by the Pythagorean
theorem A =
Ax 2  Ay 2  Az 2 .
The unit vector a can be calculated as ,
a
A
Ax i  Ay j  Az k

A
Ax 2  Ay 2  Az 2
Vector Algebra
We saw earlier that the vector P was equal to the sum of its components, Px i and
Py j. There are any number of ways to move from the origin to the position (X,Y)
where the tip of P is located. In the diagram below we will move to (X,Y) using two
other vectors, Q and R.
Now we can write P = Q + R and as before,
P = Px i + Py j . The only thing these vectors have
in common are the reference axes. Because these axes
are perpendicular to each other we can use them as the
basis for operations like addition and subtraction.
Y
R
P
Q
X
X
By resolving each vector into it components
the process of adding Q and R becomes clear.
Qx i + Rx i = Px i and Qy j + Ry j = Py j therefore
P = (Qx+ Rx) i + (Qy + Ry) j. Likewise we could
write an expression Q = P – R and show that
Q = (Px – Rx) i + (Py –Ry) j .
Perhaps a numerical example is in order.
Suppose P extends from the origin to the
point (6,7), Q to the point (9,3) and R from the
point (9,3) to the point (6,7). What are the unit
vector representations of P, Q, and R?
The components of these vectors are easy to
calculate, simply subtract the location of the tail of
the vector from the location of the tip of the vector.
Y
R
Ry j
P
Q
Qy j
Rx i
Qx i
X
Px i
Px = (6 - 0), Py = (7 - 0) , so the vector P can be expressed as P = 6 i + 7 j.
Likewise Q = 9 i + 3 j, and R = - 3 i + 4 j.
So P = Q + R = (9 + (-3)) i + (3 + 4) j = 6 i + 7 j and
Q = P – R = (6 – (-3)) i + (7 – 4) j = 9 i + 3 j .
While a vector can be multiplied or divided by a scalar there is no operation for the
division of a vector by a vector, and multiplication of a vector by a vector takes
place in two different forms called the dot product and the cross product.
Vector Multiplication
The Dot Product ( Scalar Product)
The Dot product is one way of multiplying two vectors. The outcome of this
operation is to produce a scalar quantity whose meaning varies depending on the
context of the operation. The dot product may be used to find the magnitude of a
vector, or to find the projection of a vector along some axis, or to confirm the
orthogonality of two vectors. At this point it is important to remember that this
operation is an operates on vectors to produce a scalar.
A$B=C
scalar
vector
vector
Symbol for Dot Product
The process of calculating the dot product requires a knowledge of the orientation
of the two vectors in question with respect to one another.
Py j
One can picture the process of calculating the Dot product as arranging the vectors
to be tail to tail somewhere space separated by an angle .
The calculation of their dot product is defined as
Multiplying the magnitudes of the two vectors along with
A
the cosine of the angle between them.
angle between vectors
A$B = A B cos θ = C

scalar result
magnitude
magnitude
B
A = Axi+ Ayj + Az k , B = Bxi + Byj + Bzk
Note: i$i = 1, j$j = 1,k$k = 1, and all other combinations (e.g., i$j ) are equal to
zero. In terms of the components of vectors, the dot product is
A$B = AxBx( i $ i) + AyBy( j $ j ) + AzBz( k $ k)
= AxBx + AyBy + AzBz.
Note that the dot product of A and B produces a scalar, and that A$B = B$A.
The Magnitude of a Vector:
We have
A$A = A2cosθ = A2.