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TMAT 103
Chapter 11
Vectors (§11.5 - §11.7)
TMAT 103
§11.5
Addition of Vectors: Graphical Methods
§11.5 – Addition of Vectors:
Graphical Methods
• Scalar
– Quantity that is described by magnitude
• Temperature, weight, time, etc.
• Vectors
– Quantity that is described by magnitude and
direction
• Force, velocity (i.e. wind direction and speed), etc.
§11.5 – Addition of Vectors:
Graphical Methods
• Initial Point
• Terminal Point
• Vector from A to B (vector AB)
§11.5 – Addition of Vectors:
Graphical Methods
• Arrows are often used to indicate direction
§11.5 – Addition of Vectors:
Graphical Methods
• Equal vectors – same magnitude and direction
• Opposite vectors – same magnitude, opposite
direction
§11.5 – Addition of Vectors:
Graphical Methods
• Standard position – initial point at origin
§11.5 – Addition of Vectors:
Graphical Methods
• Resultant – sum of two or more vectors
• How do we find resultant?
– Parallelogram Method to add vectors
– Vector Triangle Method
§11.5 – Addition of Vectors:
Graphical Methods
• The Parallelogram Method for adding two
vectors
§11.5 – Addition of Vectors:
Graphical Methods
• The Vector Triangle method for adding two
vectors
§11.5 – Addition of Vectors:
Graphical Methods
• The Vector Triangle method is useful when
adding more than 2 vectors
§11.5 – Addition of Vectors:
Graphical Methods
• Subtracting one vector from another
– Adding the opposite
§11.5 – Addition of Vectors:
Graphical Methods
• Examples
– Find the sum of the following 2 vectors
v = 7.2 miles at 33°
w = 5.7 miles at 61°
– An airplane is flying 200 mph heading 30° west
of north. The wind is blowing due north at 15
mph. What is the true direction and speed of
the airplane (with respect to the ground)?
TMAT 103
§11.6
Addition of Vectors: Trigonometric Methods
§11.6 – Addition of Vectors:
Trigonometric Methods
• Example:
– Use trigonometry to find the sum of the
following vectors:
v = 19.5 km due west
w = 45.0 km due north
§11.6 – Addition of Vectors:
Trigonometric Methods
• An airplane is flying 250 mph heading west.
The wind is blowing out of the north at 17
mph. What is the true direction and speed
of the airplane (with respect to the ground)?
TMAT 103
§11.7
Vector Components
§11.7 – Vector Components
• Components of a vector
– When 2 vectors are added, they are called
components of the resultant
• Special components
– Horizontal
– Vertical
§11.7 – Vector Components
• Horizontal and vertical components
§11.7 – Vector Components
If two vectors v1
and v2 add to a
resultant vector v,
then v1 and v2 are
components of v
Figure 11.49 Vectors v1 and v2 as well as u1 and u2 are components of
vector V. Vectors Vx and Vy are the horizontal and vertical components
respectively of vector v
§11.7 – Vector Components
• Horizontal and vertical components can be
found by the following formulas:
– vx = |v|cos
– vy = |v|sin
§11.7 – Vector Components
• Examples:
– Find the horizontal and vertical components of
the following vector: 30 mph at 38º
– Find the horizontal and vertical components of
the following vector: 72 ft/sec at 127º
§11.7 – Vector Components
• Examples:
– The landscaper is exerting a 50 lb. force on the handle
of the mower which is at an angle of 40° with the
ground. What is the net horizontal component of the
force pushing the mower ahead?
§11.7 – Vector Components
• Finding a vector v when its components are
known:
2
2
|
v
|

|
v
|

|
v
|
– The magnitude:
x
y
– The direction:
tan  
| vy |
| vx |
§11.7 – Vector Components
• Examples
– Find v if vx = 40 ft/sec and vy = 27 ft/sec
– Find v if vx = 10560 ft/hr and vy = 3 mph
§11.7 – Vector Components
• The impedance of a series circuit containing a resistance
and an inductance can be represented as follows. Here  is
the phase angle indicating the amount the current lags
behind the voltage.
§11.7 – Vector Components
• Example
– If the resistance is 55 and the inductive
reactance is 27, find the magnitude and
direction of the impedance.
§11.7 – Vector Components
• The impedance of a series circuit containing a resistance
and an capacitance can be represented as follows. Here 
is the phase angle indicating the amount the voltage lags
behind the current.
§11.7 – Vector Components
• Example
– If the impedance is 70 and  = 35°, find the
resistance and the capacitive reactance.