PHYS 3323 Lesson 1
Vector Algebra
A.
Freshman Definition
A physical quantity completely described by a magnitude (nonnegative number and units) and a direction.
B.
Notation
1.
2-D Polar Form

A  5.00 m  30 
Magnitude
2.
Angle
3-D Directional Cosine Form

A  6.00 N cos( 30 ) , cos( 27 ) , cos( 60 )

Magnitude

Directional Cosines
The directional cosine form that is popular in engineering classes is
actually a hybrid of the polar and Cartesian representations of a
vector. It also provides a clue as to one of the powerful uses of the dot
product.
C.
Graphical (Polar) Representation
A vector can be graphically represented as an arrow that points in
the direction of the vector and whose length gives the magnitude of
the vector.

A  4.00 m  60
D.
Graphical Addition of Vectors


To graphically add two vectors A and B ,
1.

Draw vector A starting at the origin.
2.


Draw vector B starting at the arrow tip of vector A .
3.


Draw the resultant vector C starting
at
the
origin
(tail
of
vector
)
A

and ending at the tip of vector B .
EXAMPLE: 



Find C  A  B graphically for the vectors A and B shown below.

A

B
SOLUTION:

B

A
  
CA B
Sailors used graphical addition of vectors to navigate the oceans
hundreds of years before the development of the field of vector
mathematics in the 1800’s. However, today graphical addition of
vectors is more useful for demonstrating properties of vectors and
providing visual insights than for doing computations.
EXAMPLES:
Use graphical addition of vectors to demonstrate the following
properties:
1.
2.
The result of vector addition is a vector.
(See previous examples result)
   
The addition of vectors is commutative ( A  B  B  A )
 
We calculate B  A for our previous example as
 
B A

A

B
 
And see that the resultant vector is identical to A  B .
3.
A vector can be represented as the sum of vectors pointing along
coordinate axis.
We demonstrate this property for the x and y axis.

A

Ax

Ay
 

Thus, we have A  A x  A y . We will shortly use this property to
develop the more useful component form of a vector.
The usefulness of graphical addition for making computations is
limited since:
1.
It is difficult to handle vectors in more than 2 dimensions.
2.
The precision of the resultant vector is limited by one’s drawing
ability.
E.
Unit Vectors
1.
Definition
A unit vector is a vector whose magnitude is 1.
The purpose of a unit vector is to provide direction!! We will use unit
vectors to setup different coordinate systems that are useful in
physics.
2.
Notation
A unit vector is denoted by a carrot, ^ , over the name of the
vector instead of an arrow.
EXAMPLE:
For the Cartesian coordinate system, we define three unit vectors.
î points in the +x direction.
ˆj points in the +y direction.
k̂ points in the +z direction.
F.
Cartesian (Component) Representation
We previously demonstrated that a vector can be represented as the
sum of other vectors pointing along the coordinate axis. Thus, a 3-D
vector can be represented in the Cartesian coordinate system as shown
below:

A

Ax

Az

Ay
 


A  A x  A y  Az


We now use our unit vector work to rewrite the vectors A x , A y , and

Az .

A x  A x î

A y  A y ĵ

A z  A z k̂
Ax, Ay, and Az are called the components of the vector and are scalars.
It is important to realize that a component is not generally the
magnitude of the component vector since the component may be
negative due to the definition of the unit vector.
Thus,
a 3-D vector can be represented in Cartesian coordinates as

A  A x î  A y ĵ  A z k̂ .
The component representation of a vector is extremely powerful for
making computations and for generalizing the concept of vectors to
dimensions greater than three. Unfortunately, experimental data is rarely
obtained in this form so you must learn to convert between the various
representations of a vector.
Since many problems involve spheres, cylinders and other shapes that are
difficult to represent in Cartesian coordinates, it is necessary to be able to
write vectors in component form for other coordinate systems. The key
thing to remember about component form is that you can represent any
vector provided you know the unit vectors for the coordinate system and
can determine the components of the vector. We will return to the
problem of converting vectors between coordinate systems in a later
lesson.
G.
Vector Addition in Cartesian Form

Consider the addition of two 2-D vectors, A  A î  A ĵ and
x
y

B  Bx î  B y ĵ .
By
  
CA B
Cy
Ay
Ax
Bx
Cx
From the drawing
we see that the components of the resultant

2-D vector C are given by
C x  A x  Bx
Cy  A y  By
For a 3-D vectors in Cartesian coordinate system, we can
expand our work to include the z-component as
C z  A z  Bz
These equations indicate that components along one coordinate
axis are independent of components along another coordinate
axis. This principle is used to solve projectile motion problems
in Freshman Physics. It is important to realize that this simple
form is not a general property of vectors. It is only true for
vector representations in orthogonal coordinate systems like
Cartesian, Spherical, and Cylindrical coordinates. Students
interested in additional material should consult Mathematical
Methods of Physics by Arfkin.
EXAMPLE:


Given A  5.00 î  3.00 ĵ 1.00 k̂ and B  - 2.00 î  4.00 ĵ 1.00 k̂
  
find C  A  B .
SOLUTION:

C  5.00  2.00 î  (3.00  4.00) ĵ  (1.00 -1.00) k̂

C  3.00 î  7.00 ĵ  0.00 k̂
NOTE:
Two vectors must be of the same type of physical quantity
(implies same units) in order to be added!
I.
Multiplication of a Vector by a Scalar

When
we
multiply
a
vector
by a scalar k, we are adding the vector
A

A to itself k times.
   

k A  A  A  A  ..... A
1.
Cartesian Form
From our previous work with addition in Cartesian representation, we
see that each component is added together k times. For a 2-D vector
we have:



k A  A  A  A  ..... A î   A  A  A  ..... A  ĵ
x
x
x
x
y
y
y
 y
Thus, we have the final result that

k A  k A x î  k A y ĵ
Multiply each component of the vector by the scalar k to get the components
of the new vector.
EXAMPLE:

Given A  7.00 m/s 2 î  3.00 m/s 2 ĵ  2.00 m/s 2 k̂ and M 3.00 kg ,


find F  M A .
SOLUTION:

F  (3.00 kg)(7.00 m/s 2 ) î  (3.00 kg)(3.00 m/s 2 ) ĵ  (3.00 kg)( 2.00 m/s 2 ) k̂

F  (21.0 kg m/s 2 ) î  (9.00 kg m/s 2 ) ĵ  ( 6.00 kg m/s 2 ) k̂

F  (21.0 î  9.00 ĵ  6.00 k̂ ) Newtons
2.
Geometrical (Polar Form)
To describe the resultant vector, we must answer two questions. First,
how does multiplication of a vector by a scalar effect the vectors
magnitude. Secondly, how does the operation effect the direct of the
vector.
a.
Magnitude
The magnitude of the resultant is the absolute value of the scalar times the
magnitude of the original vector.
Thus we have three specific cases:
1. If k  1then the magnitude (length) of the resultant is larger than
the magnitude (length) of the initial vector.
2. If k  1 then the magnitude (length) of the resultant is smaller than
the magnitude (length) of the initial vector.
3. If k 1 then the magnitude (length) of the resultant is the same as
the magnitude (length) of the initial vector.
b.
Direction
If the scalar is negative then the resultant vector is rotated 180 degrees
with respect to the initial vector; otherwise both vectors point in the
same direction.
EXAMPLE:



Given A  4.00 m  30 find D   2.00 A
SOLUTION:

D  (  2.00 ) ( 4.00 m ) 30   8.00 m 30  8.00 m 30  180

D  8.00 m  210 
The resultant vector has a magnitude that is four times that of the
initial vector and points in the opposite direction.
J.
Polar to Cartesian Representation Conversions
In order to solve physics problems, we generally must BVIC (Break
Vectors In To Components).
To BVIC, you just need to use simple trigonometry. The key to
obtaining each component is to multiply the magnitude of the vector
by the appropriate trigonometric function and to add a minus sign
when necessary to obtain the correct direction.
To obtain the side opposite of an angle you use the sine function.
To obtain the side adjacent to an angle you use the cosine function.
EXAMPLE:

Find the Cartesian components for the vector A shown below:
40
7.00 m
SOLUTION:
A x   A sin(40  )   7.00 m sin(40  )   4.50 m
A y   A cos(40 )   ( 7.00 m) cos(40 )  5.36 m
K.
Scalar (Dot) Product
There are two forms for the multiplication of two vectors. The scalar
(dot) product combines two vectors and produces a scalar.
1.
Notation
 
CAB
2.
Definition of the Scalar Product
The scalar product of two vectors is equal to the product of the
magnitudes of the two vectors times the cosine of the smallest angle
between the vectors.
C A B cos(θ )
The definition of the scalar product is not convenient for performing
calculations if one of the vectors is in component form. Thus, we will
use the definition of the scalar product to develop other more useful
forms of the scalar product.
3.
Geometrical Interpretation of the Scalar Product
We begin by rewriting our definition of the scalar product in the form:
C A B cos( θ )
In the diagram below,
 we see that the first term is the magnitude (total
length) of vector A while the second term is the projection


(component) of vector B in the direction of vector A .

B


A
B cos(θ )
We could also rewrite our definition of the scalar product in the form:
C B A cos( θ )
In the diagram below,
we see that the first term is the magnitude (total

length) of vector B while
 the second term is the projection

(component) of vector A in the direction of vector B .
A cos( θ )

B


A
Thus, we can summarize our results as follows:
The scalar product of two vectors is equal to the magnitude (length) of the
first vector times the component (fraction) of the second vector that lies
along the direction of the first vector.
Thus, in physics the scalar product arises when we are looking for the
projection of one vector upon another. Only the component of the
force in the direction of a body’s displacement does work on the body.
The component of the electric field along the direction normal to a
surface contributes to the electric flux.
A very useful application of this principle is the scalar product of a vector
with a unit vector. In this case, the scalar product provides the component of
the vector along the unit vector’s coordinate axis. This property will be
useful later in the course for converting between different coordinate
systems.
EXAMPLE:
Write the scalar product equation that
 can be used to find the xcomponent of a particular vector A .
SOLUTION:

A x  A  î
4.
Orthogonality

From the definition of the scalar product, it follows that vectors A and

B are perpendicular if and only if their scalar product is zero.
 
AB0
5.
Magnitude of a Vector
From the definition
of the scalar product, it follows that the magnitude

of vector A can be found by taking the square root of the scalar product
of the vector with itself
 
A  A  A A cos(0)  A 2  A
6.
Creating a Unit Vector

The unit vector that points along the direction of an arbitrary vector A can
be found by


A
A
    
A
AA
7.
Scalar Product in Cartesian Form
Our previous work on the scalar product is not convenient for finding
the scalar product between two vectors in Cartesian form as shown
below:


A  A x î  A y ĵ  A z k̂ and B  B x î  B y ĵ  Bz k̂ .
We start by writing out the scalar product of the two vectors and
performing some algebra.
C   A î  A ĵ  A k̂    B î  B ĵ  B k̂ 
y
z   x
y
z 
 x
C  A x B x  î  î   A x B y  î  ĵ  A x Bz  î  k̂   A y B x  ĵ  î   A y B y  ĵ  ĵ









 A y Bz  ĵ  k̂   A z B x  k̂  î   A z B y  k̂  ĵ  A z Bz  k̂  k̂ 








Each of the terms in parenthesis represents the cosine of an angle
between two unit vectors. Since the Cartesian coordinate system is an
orthogonal coordinate system each of these terms is either zero or one.
This greatly simplifies our final result as the cross terms are
eliminated.
C  A x B x  A y B y  A z Bz
Our algebraic approach is too cumbersome for more complicated
coordinate systems and can be greatly simplified using the Einstein
summation convention. Students wanting more information should
refer to pg 15 of Electromagnetism by Pollack and Stump and
Chapters 1 and 2 of Mathematical Methods for Physicists – 4th Edition
by Arfkin and Weber.

EXAMPLE:

A
Find the scalar product of the vectors  3.00 î  2.00 ĵ  4.00 k̂ and

B   2.00 î 1.00 ĵ  3.00 k̂ .
SOLUTION:
C  (3.00)(-2.00)  (2.00)(1.00 )  (- 4.00)(-3.00)  8.00
L.
Vector (Cross) Product
The vector (cross) product combines two vectors and produces a
vector.
1.
Notation
  
C  A B
2.
Definition of the Scalar Product
The vector product of two vectors is a vector whose magnitude is equal
to the product of the magnitudes of the two vectors times the sine of the
smallest angle between the vectors. The resultant vector’s direction is
perpendicular to the other two vectors and given by the Right Hand
Rule.
C A B sin( θ )

C
B

Bsin ( θ )

A
The definition of the cross product is not convenient for performing
calculations if one of the vectors is in component form or if the
vectors are three dimensional vectors.
3.
Geometrical Interpretation of the Vector Product
Looking at the figure above and following the procedure that we used
for the scalar product, we can develop a geometrical interpretation for
the magnitude of the vector cross product.
The magnitude of the vector cross product of two vectors is equal to the
magnitude (length) of the first vector times the component (fraction) of the
second vector that lies perpendicular to the direction of the first vector. In fact
the vector’s magnitude is equal to the area of the parallelogram formed by the
two initial vectors
B
A
Thus, in physics the vector product arises when we are looking for the
projection of one vector that is perpendicular to another. Only the
component of the force that is perpendicular to the lever arm produces
a torque. Thus, the cross product arises in situations where
rotation/circulation is being considered.
4.
Cartesian Form
When dealing with two 3-D vectors in Cartesian form, it is easier to
calculate the vector product using the determinant of matrices or
equivalently cyclic permutations. We will first develop the final
equation using algebra and the definition of the vector product.

C   A î  A ĵ  A k̂    B î  B ĵ  B k̂ 
y
z   x
y
z 
 x

C  A x B x  î  î   A x B y  î  ĵ  A x Bz  î  k̂   A y B x  ĵ  î   A y B y  ĵ  ĵ









 A y Bz  ĵ  k̂   A z B x  k̂  î   A z B y  k̂  ĵ  A z Bz  k̂  k̂ 








Using the definition of the vector product, we have the following
relationships:
î  î  ĵ  ĵ  k̂  k̂  0
î  ĵ  k̂ ĵ î   k̂
î  k̂   ĵ k̂  î  ĵ
ĵ k̂  î k̂  ĵ   î
Substituting these results into our equation, we obtain our final result

C  î (A y Bz  A z B y )  ĵ(A z Bx  A x Bz )  k̂ (A x B y  A y Bx )
It is possible to obtain this same formula without the algebra either by:
a.
determinant method
î

C  Det A x
Bx
b.
ĵ
k̂
Ay
Az
By
Bz
cyclic permutation method
Each term involves only the other two coordinate. We can write
our coordinates in a circle as shown below:

x
z
x
y
z
y
Starting from the unit vector, a clockwise rotation as shown on
the left gives a positive term while a counter-clockwise rotation
as shown on the right produces a negative term. The more
formal method of this important technique is shown in Afkin.
5.
Important
The vector product doesn’t commute!! It can easily be shown by
looking either at the Right Hand Rule or determinant method of
calculating vector products that the vector product is anticommutative:
 
 
A  B   B A
M.
Scalar Triple Product
1.
Notation
  
V  A  ( B C )
2.
Geometrical Interpretation
The scalar triple product of three vectors gives the volume of the
parallelepiped bounded by the three vectors as shown below.

C

B

A
Since the volume of a given parallelepiped is a constant (invariant),
the scalar triple product is independent of the order of the vectors.
3.
Scalar Triple Product in Cartesian Form
Ax
  
A  B C  Det B x
Cx


Ay
By
Cy
N.
Triple Cross Product
1.
Notation

  
A x BxC
2.
Az
Bz
Cz

BAC-CAB Rule
You will often find the following vector algebra relationship helpful
when attempting to simplify equations involving the triple cross
product.

 
 

        
A x Bx C  B A  C C A  B
O.
Advanced Physics Definitions
Our initial definition of a vector turns out to be vague and insufficient
for more advanced applications. In advanced physics, mathematical
equations and entities including vectors are defined by how they
behave when the coordinate system is transformed (changed). Since
the transformation properties may be constrained if a particular entity
is to represent a physical process, physicists use transformations as a
guide. As an example, consider the velocity of a car as seen by an
observer. If the observe rotates his coordinate axis (changes
direction), only the direction of the car changes not the car’s speed
(magnitude of the car’s velocity). This is true for other physical
phenomena (displacement, acceleration, force, linear momentum, etc).
Thus, one identifying property of a vector is that its magnitude (length) must
be invariant (unchanged) under rotation!!
1.
Vector
Continuing with this process, advanced physics courses define a vector as a
quantity with three components that transforms under rotation of the coordinate
axis in the same manner as a point (the position vector).
2.
Scalar
A scalar is a quantity that is invariant (does not change) under rotation of the
coordinate axis.
3.
This material is presented in the textbook to indicate why physicist
say that the Del operator is a vector and as an introduction to the way
theoretical physics is performed. It will not be tested or emphasized in
this course.