TMAT 103 Chapter 11 Vectors (§11.5 - §11.7). TMAT 103 §11.5 Addition of Vectors: Graphical...

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

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

If two vectors v1 and v2 add to a resultant vector v, then v1 and v2 are components of 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:– The magnitude:

– The direction:

22 |||||| yx vvv

||

||tan

x

y

v

v

§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.