Vectors
Chapter Outline
ENGINEERING PHYSICS I
Coordinate Systems
PHY 303K
Vector and Scalar Quantities
Chapter 3: Vectors
Some Properties of Vectors
Components of a Vector and Unit Vectors
Maxim Tsoi
Physics Department,
The University of Texas at Austin
http://www.ph.utexas.edu/~tsoi/303K.htm
303K: Ch.3
303K: Ch.3
Coordinate Systems
Vector and Scalar Quantities
Description of a location in space
Magnitude and Direction
• Cartesian coordinate system (rectangular coordinates)  (x, y)
• Polar coordinate system  (r, )
x  r cos 
y  r sin 
• A scalar quantity is completely specified
by a single value with an appropriate unit
y
tan  
x
and has no direction (e.g., temperature,
volume, mass, speed, time intervals)
r  x2  y2
• A vector quantity is completely
specified by a number and appropriate

units plus a direction (e.g., velocity,
A, A , A, |A|
displacement)
Example 3.1
303K: Ch.3
303K: Ch.3
Some Properties of Vectors
Some Properties of Vectors
Equality of two vectors
• Two vectors A and B are equal if they have the same magnitude
and point in the same direction (A=B, A||B)
Laws of vector addition
• The rules for adding vectors are conveniently described by graphical methods
• To add vector B to vector A:
• Draw B with its tail starting from the
tip of A
• The vector R=A+B is the vector
drawn from the tail of A to the tip of B
Total displacement is the vector sum
of the individual displacements
303K: Ch.3
303K: Ch.3
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Some Properties of Vectors
Some Properties of Vectors
Adding more than two vectors
• The rules for adding vectors are conveniently described by graphical methods
• To add four (N) vectors A, B, C, and D:
Commutative law of addition
• The rules for adding vectors are conveniently described by graphical methods
• When two vectors are added, the sum is
independent of the order of the addition:
• Draw 2nd (3rd, etc.) vector with its tail
starting from the tip of the 1st (2nd, etc)
• A+B=B+A
• R=A+B+C+D is the vector drawn from
the tail of the first vector to the tip of
the last vector
Total displacement is the vector sum
of the individual displacements
303K: Ch.3
Total displacement is the vector sum
of the individual displacements
303K: Ch.3
Some Properties of Vectors
Some Properties of Vectors
Associative law of addition
• The rules for adding vectors are conveniently described by graphical methods
• When three or more vectors are added, their
Negative of a vector
• The rules for adding vectors are conveniently described by graphical methods
• The negative of the vector A is defined as the
sum is independent of the way in which the
vector that when added to A gives zero for the
individual vectors are grouped together:
vector sum:
• A+(B+C)=(A+B)+C
• A+(-A)=0
• The vectors A and -A have the same
magnitude but point in opposite directions
A
-A
• All vectors must
have the same units
and type of quantity
Total displacement is the vector sum
of the individual displacements
303K: Ch.3
303K: Ch.3
Some Properties of Vectors
Some Properties of Vectors
Subtracting vectors
• The rules for adding vectors are conveniently described by graphical methods
• Make use of the definition of the negative of a
vector
• We define subtraction A-B as vector –B added
Multiplying a vector by a scalar
• The rules for adding vectors are conveniently described by graphical methods
• m – positive  the product mA is a vector that has the
same direction as A and magnitude mA
• -m – negative  the product -mA is directed opposite A
to vector A:
• A – B = A + (-B)
A
5A
303K: Ch.3
B
-½ B
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Components of a Vector and Unit Vectors
Components of a Vector and Unit Vectors
Adding vectors using their projections
Rotation of a coordinate system
• The projections of vector A along coordinate axes
• In many applications/problems it is convenient to express the components
are called the components Ax and Ay of the vector
of a vector A in a coordinate system having axes that are not horizontal
and vertical but are still perpendicular to each other
• Any vector can be completely described by its
components
• In different coordinate systems the components of the same vector must
be modified accordingly
A  Ax2  Ay2
Ax  A cos 
 Ay 

 Ax 
Ay  A sin 
y’
A=Ax+Ay
  tan 1 
 the signs of the components
depend on the angle 
x’
Ax  A cos
Ax '  A
Ay  A sin 
Ay '  0
 a vector A can be specified either
with Ax and Ay or with A and `
303K: Ch.3
303K: Ch.3
Components of a Vector and Unit Vectors
Components of a Vector and Unit Vectors
Unit vectors
Using components to add vectors
 A unit vector is a dimensionless vector having a magnitude of exactly 1
• Used to specify a given direction and have no other physical significance

 
î , ĵ , k̂  x , y , z
A  A x  A y  Ax î  Ay ĵ
r  x î  y ĵ

R  A  B  Ax î  Ay ĵ  B x î  B y ĵ   Ax  B x î  Ay  B y ĵ  Rx î  R y ĵ
• Vector quantities are often expressed in terms of unit vectors
R  Rx2  R y2 
 Ax  Bx 2  Ay  B y 2
tan  
Ry
Rx

Ay  B y
Ax  B x
î  ĵ  k̂  1
303K: Ch.3
303K: Ch.3
Components of a Vector and Unit Vectors
SUMMARY
Vector components in 3D
Vectors
• SCALAR quantities  have only a numerical value; no direction
A  Ax î  Ay ĵ  Az k̂
• VECTOR quantities  have both magnitude (>0) and direction; obey
B  B x î  B y ĵ  Bz k̂
the laws of vector addition
• R=A+B is the vector drawn from the tail of A to the tip of B
R   Ax  B x î  Ay  B y ĵ   Az  Bz k̂
R  Rx2  R 2y  Rz2
cos  x 
Rx
R
cos  y 
Ry
R
cos  z 
• The x (y) component Ax (Ay) of the vector A is equal to the
projection of A along the x (y) axis of a coordinate system
Rz
R
• A vector A can be expressed via its components
• Where
î
and
ĵ
are unit vectors (
A  Ax î  Ay ĵ
î  ĵ  1 ) in x and y
directions
303K: Ch.3
303K: Ch.3
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