Chapter 3: Vectors

3.1 Coordinate Systems

3.2 Vector and Scalar Quantities
3.3 Some Properties of Vectors
3.4 Components of a Vector and Unit
Vectors

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 3.1 Coordinate Systems
„ Used to describe the position of a point
in space
„ Coordinate system consists of
„ a fixed reference point called the origin
„ specific axes with scales and labels
„ instructions on how to label a point relative
to the origin and the axes

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 Cartesian Coordinate System
„ Also called
rectangular
coordinate system
„ x - and y - axes
intersect at the
origin
„ Points are labeled
(x,y)

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 Polar Coordinate System
„ Origin and reference
line are noted
„ Point is distance r
from the origin in
the direction of
angle θ, ccw from
reference line
„ Points are labeled
(r,θ)
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 Polar to Cartesian Coordinates
„ Based on
forming a right
triangle from r
and θ
„ x = r cos θ
„ y = r sin θ

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 Cartesian to Polar Coordinates
„ r is the hypotenuse and
θ an angle
y
tan θ =
x
r = x2 + y2

„ θ must be ccw from

positive x axis for these
equations to be valid

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 Example 3.1
„ The Cartesian coordinates of
a point in the xy plane are
(x,y) = (-3.50, -2.50) m, as
shown in the figure. Find
the polar coordinates of this
point.

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 Example 3.1
„ The Cartesian coordinates of a
point in the xy plane are (x,y)
= (-3.50, -2.50) m, as shown
in the figure. Find the polar
coordinates of this point.

„ Solution: From Equation 3.4,

r = x 2 + y 2 = ( −3.50 m) 2 + ( −2.50 m) 2 = 4.30 m

and from Equation 3.3,
y −2.50 m
tan θ = = = 0.714
x −3.50 m
θ = 216°
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 3.2 Vectors and Scalars
„ A scalar quantity is completely
specified by a single value with an
appropriate unit and has no direction.
„ A vector quantity is completely
described by a number and appropriate
units plus a direction.

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 Vector Notation
r
„ When handwritten, use an arrow: A
„ When printed, will be in bold print: A
„ When dealing with just the magnitude of a
vector in print, an italic letter will be used: A
or |A|
„ The magnitude of the vector has physical
units
„ The magnitude of a vector is always a
positive number

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 Vector Example
„ A particle travels from A
to B along the path
shown by the dotted
red line
„ This is the distance
traveled and is a scalar
„ The displacement is
the solid line from A to
B
„ The displacement is
independent of the path
taken between the two
points
„ Displacement is a vector
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 3.3 Some Properties of Vectors
Equality of Two Vectors
„ Two vectors are
equal if they have
the same magnitude
and the same
direction
„ A = B if A = B and
they point along
parallel lines
„ All of the vectors
shown are equal
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 Adding Vectors
„ When adding vectors, their directions
must be taken into account
„ Units must be the same
„ Graphical Methods
„ Use scale drawings
„ Algebraic Methods
„ More convenient

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 Adding Vectors Graphically
„ Choose a scale
„ Draw the first vector with the appropriate
length and in the direction specified, with
respect to a coordinate system
„ Draw the next vector with the appropriate
length and in the direction specified, with
respect to a coordinate system whose origin
is the end of vector A and parallel to the
coordinate system used for A
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 Adding Vectors Graphically, 2
„ Continue drawing the
vectors “tip-to-tail”
„ The resultant is drawn
from the origin of A to
the end of the last
vector
„ Measure the length of R
and its angle
„ Use the scale factor to
convert length to actual
magnitude

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 Adding Vectors Graphically, 3
„ When you have
many vectors, just
keep repeating the
process until all are
included
„ The resultant is still
drawn from the
origin of the first
vector to the end of
the last vector

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 Adding Vectors, Rules
„ When two vectors
are added, the sum
is independent of
the order of the
addition.
„ This is the
Commutative Law
of Addition
„ A+B=B+A

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 Adding Vectors, Rules, 2
„ When adding three or more vectors, their sum is
independent of the way in which the individual
vectors are grouped
„ This is called the Associative Property of Addition
„ (A + B) + C = A + (B + C)

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 Adding Vectors, Rules, 3
„ When adding vectors, all of the vectors
must have the same units
„ All of the vectors must be of the same
type of quantity
„ For example, you cannot add a
displacement to a velocity

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 Negative of a Vector
„ The negative of a vector is defined as
the vector that, when added to the
original vector, gives a resultant of zero
„ Represented as –A
„ A + (-A) = 0
„ The negative of the vector will have the
same magnitude, but point in the
opposite direction

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 Subtracting Vectors
„ Special case of
vector addition
„ If A – B, then use
A+(-B)
„ Continue with
standard vector
addition procedure

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 Multiplying or Dividing a
Vector by a Scalar
„ The result of the multiplication or division is a
vector
„ The magnitude of the vector is multiplied or
divided by the scalar
„ If the scalar is positive, the direction of the
result is the same as the original vector
„ If the scalar is negative, the direction of the
result is opposite that of the original vector

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 3.4 Components of a Vector
„ A component is a
part
„ It is useful to use
rectangular
components
„ These are the
projections of the
vector along the x-
and y-axes

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 Vector Component Terminology
„ Ax and Ay are the component vectors
of A
„ They are vectors and follow all the rules for
vectors
„ Ax and Ay are scalars, and will be
referred to as the components of A

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 Components of a Vector, 2
„ The x-component of a vector is the
projection along the x-axis
Ax = A cosθ
„ The y-component of a vector is the
projection along the y-axis
Ay = A sin θ
„ Then, A = A x + A y

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 Components of a Vector, 3
„ The y-component is
moved to the end of
the x-component
„ This is due to the
fact that any vector
can be moved
parallel to itself
without being
affected
„ This completes the
triangle
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 Components of a Vector, 4
„ The previous equations are valid only if θ is
measured with respect to the x-axis
„ The components are the legs of the right
triangle whose hypotenuse is A
Ay
A= A +A
2
x
2
y and θ = tan −1

Ax
„ May still have to find θ with respect to the positive
x-axis

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 Components of a Vector, 5
„ The components can
be positive or
negative and will
have the same units
as the original
vector
„ The signs of the
components will
depend on the angle

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 3.4 Unit Vectors
„ A unit vector is a dimensionless vector
with a magnitude of exactly 1.
„ Unit vectors are used to specify a
direction and have no other physical
significance

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 Unit Vectors, cont.
„ The symbols
ˆi , ˆj, and kˆ
represent unit vectors
in Cartesian coordinate
system
„ Form a set of mutually
perpendicular vectors
(i.e. orthogonal vectors)

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 Unit Vectors in Vector Notation
„ Ax is the same as Ax î
and Ay is the same
as Ay ĵ ,etc.
„ The complete vector
can be expressed as
A = Ax ˆi + Ay ˆj + Az kˆ

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 Adding Vectors Using Unit Vectors
„ Using R = A + B
„ ( ) (
Then R = Ax ˆi + Ay ˆj + Bx ˆi + B y ˆj )
(
= ( Ax + Bx )ˆi + Ay + B y ˆj )
= Rx ˆi + R y ˆj
„ and so Rx = Ax + Bx and Ry = Ay + By
Ry
R= R +R2
x
2
y θ = tan −1

Rx
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 Adding Vectors with Unit Vectors

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 Trig Function Warning
„ The component equations (Ax = A cosθ
and Ay = A sinθ) apply only when the
angle is measured ccw with respect to
the positive x-axis.
„ The resultant angle (tanθ = Ay /Ax)
gives the angle with respect to the
positive x-axis.

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 Adding Vectors Using Unit
Vectors – Three Directions
„ Using R = A + B
( ) (
R = Ax ˆi + Ay ˆj + Az kˆ + Bx ˆi + B y ˆj + Bz kˆ )
( )
= ( Ax + Bx )ˆi + Ay + B y ˆj + ( Az + Bz )kˆ
= Rx ˆi + R y ˆj + Rz kˆ
„ Rx = Ax + Bx , Ry = Ay + By and Rz = Az + Bz
Rx
R= R +R +R2
x
2
y
2
z
θ x = cos −1
etc.
R
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 Example 3.5: Taking a Hike
„ A hiker begins a trip by first walking 25.0 km
southeast from her car. She stops and sets
up her tent for the night. On the second day,
she walks 40.0 km in a direction 60.0° north
of east, at which point she discovers a forest
ranger’s tower.

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 Example 3.5
„ (A) Determine the components
of the hiker’s displacement for
each day.

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 Example 3.5

Ax = A cos( −45.0°) = (25.0 km)(0.707) = 17.7 km

Ay = A sin( −45.0°) = (25.0 km)( −0.707) = −17.7 km

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 Example 3.5

Bx = B cos 60.0° = (40.0 km)(0.500) = 20.0 km

B y = B sin 60.0° = (40.0 km)(0.866) = 34.6 km

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 Example 3.5

R=A+B:
Rx = Ax + Bx = 17.7 km + 20.0 km = 37.7 km
Ry = Ay + By = -17.7 km + 34.6 km = 16.9 km
In unit-vector form, we can write the total displacement as
R = (37.7 î + 16.9 ĵ) km

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 Example 3.5
„ Using Equations 3.16 and 3.17, we
find that the vector R has a
magnitude of 41.3 km and is
directed 24.1° north of east.

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