VECTORS The study of vectors is closely related to the study of such physical properties as force,...

VECTORS

The study of vectors is closely related to the study of such

physical properties as force, motion, velocity, and other related

topics.

Vectors allow us to model certain characteristics of these

phenomena with numbers that tell us their magnitude and

direction.

SCALAR QUATITIES:

Measurements involving such things as time, area, volume,

energy, and temperature are called scalar measurements

because each can be described adequately using their

magnitude alone (with the appropriate units).

27 ft3 adequately describes the volume of a cube with side 3

ft. 980 F adequately describes the temperature of a person.

27ft3 and 980 F are called scalars.

Some properties such as force, velocity, and displacement

require both magnitude and direction to be described

completely. These quantities are called vector quantities.

Example: You are driving due north at 45 miles per hour. The magnitude is the speed, 45 miles per hour. The direction of motion is due north.

NOTATION AND GEOMETRY OF VECTORS

Two airplanes travel at 400 mph on a parallel course and in the same direction. This situation can be modeled using directed line segments.

A

C

B

A

D

The directed segments are drawn parallel with arrowheads pointing the same way to indicate direction of flight, while making them the same length indicates that the velocities are the same. The length of the vector models the magnitude of the velocity, while the arrowhead indicates the direction of travel.

Vectors are named using the initial and terminal points that define them as inOr with a bold, small case letter such as u or v. We may also write them as .

AB and CD��

u or v

The magnitude of the directed line segment is its length.

We indicate this by . is the distance from point P

to point Q.

P

Q

PQ��

PQ��

PQ��

Reference Angle θr :For any angle θ in standard position, the acute angle θr formed by the terminal side and the x-axis is called the reference angle for θ.

Find the reference angle, θr , for each of the following angles.a) θ = 3450 b) θ = -1350 c) 5800

a) θr = 150 b) θr = 450 c) θr = 400

Example

x

y

u

v

(2,3)

( 2, 1)

( 4, 4)

Show that u=v.

Vectors that are equal have the same magnitude and direction.Use the distance formula to show that have the same magnitude.

u and v

2 22 1 2 1d = (x ) ( )x y y

2 2[0 ( 3)] [3 ( 3)]u

13

2 2(3 0) (6 0)v

13

Thus and have the same magnitude: u v u v��

One way to show that have the same direction is to find the slopes of the lines on which they lie.

u and v

2 1

2 1

y ym

x x

Verify to show that each vector has a slope of 3/2.

POSITION VECTORS

For a vector v with initial point (x1, y1) and terminal point (x2, y2), the position vector for v is

an equivalent vector with initial point (0,0) and terminal point (x2 – x1, y2 – y1).

= ,v 2 1 2 1x - x , y -y

The vector in component form is denoted as ,

where a is the horizontal component and b is the vertical component.

,a b

Find the position vector for vector u and graph it.

= -2 - (-5), 3 - (-4) = v 3, 7

The position vector for u is:

The position vector v has (0,0) as its initial point, and (3, 7) as its terminal point.

u

(-5, -4)

(-2, 3)

3

7

(3, 7)

For a position vector v = ‹a, b› shown below at left and angle θr, observe the following:

a

b

x

y

θr

v

The magnitude of vector v = 2 2a b

rv cosθa

rv sinθb

rtanθb

a

1rθ tan

b

a

rsinθv

bvertical component:

rcosθv

ahorizontal component:

Finding the Magnitude and Direction Angle of a Vector

1 2Example: For v 2.5, 6 and v 3 3, 3 , ��

a. Find their magnitudes.

b. Graph each vector and name the quadrant where located.

c. Find the angle θ for each vector (round to tenths of a degree).

2 2

1a. v 2.5 6 6.25 36 42.2 6 55 . ��

22

2b. v 3 3 3 27 9 636 ��

11 r

6c. For v : θ tan

2.5

1 0tan 2. 44 67.

12 r

3d. For v : θ tan

3 3

01 3tan

330

is located in QIII. Why?

1v 2.5, 6 ��

2v 3 3, 3 in QI.��

300

3 3, 3

2.5, 6

Find the Horizontal and Vertical Components of a Vector.

0

The vector , is in QIII, has a magnitude of = 21, and forms

an angle of 25 with the negative x-axis. Graph the vector and find its

horizontal and vertical components.

a bv = v��

x

y

250

-19

-8.9

21v��

Note: θr = 25⁰, therefore θ = 1800 + 250 = 2050.

0

Horizontal Component:

= cos 21c 19os205a v��

0

Vertical Component:

sin 21sin205 8.9b b v��

Vector Addition:

Addition of vectors using the “tail – to – tip” method.Shift one vector (without changing its direction) so that its tail (initial point) is at the tip (terminal point) of the other vector.Given vectors u and v

x

y x

y

vu

Tail of v to tip of u

uuu + v

v

x

y

v

Tail of u to tip of v

u

u + v

v

v

w

Initial point of v

Terminal point of w

Vector Addition: Add vectors v and w.

��v +w

v

w

APPLICATION OF VECTORS:A common example of a vector quantity is force. Other vector quantities that appear in engineering mechanics are moment, displacement, velocity, and acceleration.

FORCE: The effect of one physical body on another physical body. The force effect between two bodies can be interpreted as a ‘push’ or ‘pull’of one of the bodies on the other body.

RESULTANT FORCE: When two or more forces are added to obtain a single force, it produces the same effect as the original system of forces. This single vector is called the sum, or the resultant force of the original system of forces.

F1

F2

FR

y

x

The resultant force, FR , is the sum of F1 and F2 .

COMPONENTS OF A FORCE:Two or more forces acting on a particle may be replaced

by a single force which has the same effect on the

particle.

Conversely, a single force F acting on a particle may be

replaced by two or more forces which, together, have the

same effect on the particle. These forces are called the

components of the original force F.

RECTANGULAR COMPONENTS OF A FORCE.

Often it is desirable to resolve a force into two components which are perpendicular to each other. In the figure below, the force F has been resolved into a component Fx and a component Fy. Fx and Fy are called rectangular components.

Fy

0

F

y

Fx

xθ

Fx = FcosθFy = Fsinθ

y

x

Ftanθ=

F

y-1

x

Fθ=tan

F

Example: A force of 800 N is exerted on a bolt A as shown. Determine the horizontal and vertical components of the force.

Note: θR = 350; What is θ ?

350

F = 800 N

A

A

Fy

Fx

F = 800 N

350θ = 1450

Fx = Fcos1450 = 800cos1450 = -655 N

Fy = Fsin1450 = 800sin145˚ = 459 N

We write F in the form F = -(655N)i + (459 N) j

Vectors in the Rectangular Coordinate System

The i and j Unit Vectors.

Vector i is the unit vector (vector of length 1) whose direction is along the positive x-axis. Vector j is the unit vector whose direction is along the positive y-axis.

y

1

1

j

ix

Vectors in the rectangular coordinate system can be represented in terms of i and j.

b

ax

0

P = (a, b)

v = ai + bj

A unit vector is defined to be a vector whose magnitude is one. In many applications, it is useful to find the unit vector that has the same direction as a given vector.

vFor any nonzero vector , the vector is the unit vector that has the

same direction as To find this vector, divide by its magnitude.

vv

v. v

��

Example: Find the unit vector in the same direction as v = 5i – 12j. Then verify that the vector has magnitude 1.

2 25 ( 12) 25 144 169 13 v��

5 12 5 12

13 13 13

v i ju j

v��

2 25 12 25 144 169( ) ( ) ( ) ( ) 113 13 169 169 16

19

Verify :

Sketch the vector and find its magnitude.v = -3i + 4j

a = -3 and b = 4.

(-3, 4)

v = 5

Example

x

y

Sketch the vector v=-2i+j and

find its magnitude.

Example

x

y

1

2

Let v be a vector from initial point P ( 2, 2)

to terminal pont P (2,3). Write v in terms

of i and j.

Operations with Vectors in Terms of i and j

Vector Subtraction:

The difference of two vectors, u – v is defined asu – v = u + (-v):

x

y

vu

x

y

u

The terminal point of u coincides with the initial point of -v

Example

If v=2i-3j and w=-i+4j find each of the following:

a. v+w

b. v-w

c. 2v

d. -3w

e. 2v-3w

Unit Vectors

Example

Find the unit vector in the same direction as v=-3i-4j.

Then verify that the vector has the magnitude of 1.

Writing a Vector in Terms of Its Magnitude and Direction

Example

0The wind is blowing at 16 mph in the direction of N60 E.

Express its velocity as a vector v in terms of i and j.

Application

Example

1 2

1

0 02

Two forces F and F , of magnitude 5 lbs and 12 lbs,

respectively act on an object. The direction of F is

N20 E and the direction of F is N75 E. Find the

magnitude and the direction of the resultant force.

Express the magnitude to the nearest hundredth of a

pound and the direction angle to the nearest tenth

of a degree.

(a)

(b)

(c)

(d)

If u=- 3i+4j, v=2i-j find 2u+v

-i+3j

4i-j

5i+3j

-4i+7j

(a)

(b)

(c)

(d)

Find the magnitude of the vector v=-3i+4j

5

6

7

8

Converting from Rectangular Coordinates to Polar Coordinates.

a) Radians.

b) Polar Coordinates.

We measure angles by determining the amount of rotation from the initial side to the terminal side. Two units of measurement for angles are degrees and radians.

A) RADIAN MEASURE

An angle whose vertex is at the center of a circleis called a central angle. A central angle intercepts the arc of the circle from the initial side to the terminal side. A positive central angle that intercepts an arc of the circle of length equal to the radius of the circle has a measure of 1 radian.

Radian Measure

r rө

How many radians are there in a circle? 6.28

How many degrees are there in 1 radian? 360

6.28 o57.3

RELATIONSHIP BETWEEN DEGREES AND RADIANS:

360˚ = 2π radians 180˚ = π radians

Degrees to radians:

o πθ =θ radians

180

180

θradians= θ degreesπ

Radians to degrees:

Convert each angle from degrees to radians.a) 30˚ b) 90˚ c) -225˚ d) 55˚

180

π ra) ad30 30 ians

6

π radb) 90 ians

2

5π- rc) -2 ad25 ians

4

55d) 55 55

180 180

0.96radians

Convert each angle in radians to degrees.

π 3πa) radians b) = - radians c) 2.5radians

3 4

180a) radians

3 3

o60

3 3 180b) - radians = -

4 4

o-135

180c) 2 radians 2

o114.6

The foundation of the polar coordinate system is a horizontal ray that extends to the right. This ray is called the polar axis. The endpoint of the polar axis is called the pole. A point P in the polar coordinate system is designated by an ordered pair of numbers (r, θ).

r

pole

polar axis

θ

P = (r, θ)r is the directed distance form the pole to point P ( positive, negative, or zero).

θ is angle from the pole to P (in degrees or radians).

0 0

0

To plot the point P( r,θ) , go a distance of

r at 0 then move θ along a circle of

radius r.

If r > 0, plot a point at that location. If r < 0, the

point is plotted on a circle of the same radius,

but 180 in the opposite direction.

Plotting Points in Polar Coordinates.

Plot each point (r, θ)

a) A(3, 450)

A b) B(-5, 1350)

Bc) C(-3, -π/6)

C

CONVERTING BETWEEN POLAR AND RECTANGULAR FORMS

CONVERTING FROM POLAR TO RECTANGULAR COORDINATES.To convert the polar coordinates (r, θ) of a point to rectangular coordinates (x, y), use the equations

x = rcosθ and y = rsinθ

Convert the polar coordinates of each point to its rectangular coordinates.a) (2, -30⁰ ) b) (-4, π/3)

a) x = rcos(-30⁰) 3

2( ) 32

2sin( 30 ) 2( 1/ 2) 1y

The rectangular coordinates of (2, 30 ) are ( 3,-1)

b) x= -4cos(π/3) = -4(1/2) = -2

y= -4 sin(π/3) = 3

4( ) 2 32

The rectangular coordinates of (-4, ) are 3

( -2, -2 3)

CONVERTING FROM RECTANGULAR TO POLAR COORDINATES:

To convert the rectangular coordinates (x, y) of a point to polar coordinates:

1)Find the quadrant in which the given point (x, y) lies.

2) Use r = 2 2 to find .x y r

3) Find by using tan and choose so that it lies in the

same quadrant as the point ( , ).

y

x

x y

Find the polar coordinates (r, θ) of the point P with r > 0 and 0 ≤ θ ≤ 2π, whose rectangular coordinates are (x, y) = ( 1, 3)

The point is in quadrant 2.

2 2( 1) ( 3) 4 2r

tanθ = 3

1

1tan 3 60r

2180 60 120 120 ( )

180 3or

The required polar coordinates are (2, 2π/3)

10.1 Three-Dimensional Coordinate Systems

Distance Formula in Three Dimensions

1 2 1 1 1 1 2 2 2 2

2 2 21 2 2 1 2 1 2 1

The distance between the points (x , , )and (x , , ) is

( x ) ( ) ( )

PP P y z P y z

PP x y y z z

Equation of a sphereAn equation of a sphere with center C(h, k, l) and radius r is (x – h)2 + (y – k)2 + (z – l)2

P(4, 9, 12)Q(4, 0, 12)

R(0, 0, 12)

VECTORS The study of vectors is closely related to the study of such physical properties as force,...

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