Vectors
•
•
•
•
Scalars and Vectors
Vector Components and Arithmetic
Vectors in 3 Dimensions
Unit vectors i, j, k
Serway and Jewett Chapter 3
Physics 1D03 - Lecture 3
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Physical quantities are classified as scalars,
vectors, etc.
Scalar : described by a real number with units
examples: mass, charge, energy . . .
Vector : described by a scalar (its magnitude) and
a direction in space
examples: displacement, velocity, force . . .
Vectors have direction, and obey
different rules of arithmetic.
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Notation
•
Scalars : ordinary or italic font (m, q, t . . .)
•
Vectors : - Boldface font (v, a, F . . .)
  
- arrow notation ( v, a, F . . .)
- underline (v, a, F . . .)
•
Pay attention to notation :
“constant v” and “constant v” mean
different things!
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Magnitude : a scalar, is the “length” of a vector.
e.g., Speed, v = |v| (a scalar), is the magnitude of velocity v
Multiplication:

A
scalar  vector = vector

3 A
2

 12 A
Later in the course, we will use two other types of multiplication:
the “dot product” , and the “cross product”.
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Vector Addition: Vector + Vector = Vector

A
  
e.g. A  B  C
Triangle Method

A

B
  
C  AB

B
Parallelogram Method

A
  
C  AB

B
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Concept Quiz
Two students are moving a refrigerator.
One
pushes with a force of 200 newtons, the other with a
force of 300 newtons. Force is a vector. The total
force they (together) exert on the refrigerator is:
a) equal to 500 newtons
b) equal to
2002  3002 newtons
c) not enough information to tell
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Concept Quiz
Two students are moving a refrigerator. One pushes
with a force of 200 newtons (in the positive direction),
the other with a force of 300 newtons in the opposite
direction. What is the net force ?
a)100N
b)-100N
c) 500N
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Coordinate Systems
In 2-D : describe a location in a plane
y
• by polar coordinates :
(x,y)
distance r and angle 
r
y
• by Cartesian coordinates :

0
x
x
distances x, y, parallel to axes
with: x=rcosθ y=rsinθ
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Components
• define the axes first
y
• v x , v y , (andv z ) are scalars
• axes don’t have to be
horizontal and vertical

v
vy
vx
• the vector and its
components form a right
triangle with the vector on
the hypotenuse
Physics 1D03 - Lecture 3
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3-D Coordinates (location in space)
We use a right-handed coordinate system with
three axes:
z
y
y
x
x
z
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x
Is this a right-handed
coordinate system?
Does it matter?
y
z
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Unit Vectors
ˆ is a vector with
A unit vector u or u
ˆ 1
magnitude 1 : u
z
(a pure number, no units)
k
Define coordinate unit vectors i, j, k
along the x, y, z axis.
j
y
i
x
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
A vector A can be written in terms of its components:




A  Ax i  Ay j  Az k

A
Ay j

A
Ay j
j
i
Ax i
Ax i
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Addition again:
If A + B = C ,

A
Ay
Ax

B
By
Bx
then:
Cx  Ax  Bx
Tail to Head
Cy  Ay  By

B

C
Cz  Az  Bz
Three scalar
equations from one
vector equation!
Cy
Bx

A
Ax
By
Ay
Cx
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The unit-vector notation leads to a simple rule for
the components of a vector sum:
  
AB C
In components (2-D for simplicity) :
 
A  B  ( Ax  Bx) i  ( Ay  By ) j  Cx i  Cy j
Eg: A=2i+4j
B=3i-5j
A+B = 5i-j
A - B = -i+9j
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Magnitude : the “length” of a vector.
Magnitude is a scalar.
e.g., Speed is the magnitude of velocity:
velocity = v ; speed = |v| = v
In terms of components:
y
| v | v x2  v y2
On the diagram,
vx = v cos 

v
vy

vx
x
vy = v sin 
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Summary
• vector quantities must be treated according to the
rules of vector arithmetic
• vectors add by the triangle rule or parallelogram rule
(geometric method)

• a vector A can be represented in terms of its
Cartesian components using the “unit vectors” i, j, k
these can be used to add vectors (algebraic
method)
  
• C  A  B if and only if:
Cx  Ax  Bx
Cy  Ay  By
Cz  Az  Bz
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