Chapter3B Vectors

Chapter 3B Chapter 3B -- VectorsVectors

A PowerPoint Presentation byA PowerPoint Presentation byA PowerPoint Presentation byA PowerPoint Presentation byA PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of PhysicsPaul E. Tippens, Professor of Physics

Southern Polytechnic State UniversitySouthern Polytechnic State University

A PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of PhysicsPaul E. Tippens, Professor of Physics

Southern Polytechnic State UniversitySouthern Polytechnic State University

2007

VectorsVectors

Surveyors use accurate measures of magnitudes and directions to create

scaled maps of large regions.

Objectives: After completing this Objectives: After completing this module, you should be able to:module, you should be able to:

Demonstrate that you meet Demonstrate that you meet mathematics mathematics expectationsexpectations: unit analysis, algebra, scientific : unit analysis, algebra, scientific notation, and rightnotation, and right--triangle trigonometry.triangle trigonometry.

Define and give examples of Define and give examples of scalarscalar and and vector vector

Demonstrate that you meet Demonstrate that you meet mathematics mathematics expectationsexpectations: unit analysis, algebra, scientific : unit analysis, algebra, scientific notation, and rightnotation, and right--triangle trigonometry.triangle trigonometry.

Define and give examples of Define and give examples of scalarscalar and and vector vector Define and give examples of Define and give examples of scalarscalar and and vector vector quantities.quantities.

Determine the Determine the componentscomponents of a given vector.of a given vector.

Find the Find the resultant resultant of two or more vectors.of two or more vectors.

Define and give examples of Define and give examples of scalarscalar and and vector vector quantities.quantities.

Determine the Determine the componentscomponents of a given vector.of a given vector.

Find the Find the resultant resultant of two or more vectors.of two or more vectors.

ExpectationsExpectations

You must be able convert units of You must be able convert units of measure for physical quantities.measure for physical quantities.

Convert 40 m/s into kilometers per hour.Convert 40 m/s into kilometers per hour.

40--- x ---------- x -------- = 144 km/h m

s

1 km

1000 m

3600 s

1 h

Expectations (Continued):Expectations (Continued):

College algebra and simple formula College algebra and simple formula manipulation are assumed.manipulation are assumed.

Example:0 fv v

x t+

= Solve for vExample:0

2f

x t

=

Solve for vo

02fv t x

vt

=

Expectations (Continued)Expectations (Continued)

You must be able to work in scientific You must be able to work in scientific notation.notation.

Evaluate the following:Evaluate the following:

(6.67 x 10-11)(4 x 10-3)(2)

(8.77 x 10-3)2F = -------- = ------------

Gmm

r2

F = 6.94 x 10-9 N = 6.94 nN


You must be familiar with SI prefixesYou must be familiar with SI prefixes

The meter (m) 1 m = 1 x 100 m

1 Gm = 1 x 109 m 1 nm = 1 x 10-9 m

1 Mm = 1 x 106 m 1 m = 1 x 10-6 m

1 km = 1 x 103 m 1 mm = 1 x 10-3 m


You must have mastered rightYou must have mastered right--triangle triangle trigonometry. trigonometry.

y = R sin sin yR

=y

x

R

x = R cos x = R cos

R

cosx

R =

tanyx

= R2 = x2 + y2

Mathematics ReviewMathematics Review

If you feel you need to If you feel you need to brush up on your brush up on your mathematics skills, try mathematics skills, try the tutorial from Chap. the tutorial from Chap. 2 on Mathematics. Trig 2 on Mathematics. Trig

Select Chap. 2 from the On-Line Learning Center in TippensStudent Edition

2 on Mathematics. Trig 2 on Mathematics. Trig is reviewed along with is reviewed along with vectors in this module.vectors in this module.

Physics is the Science of Physics is the Science of MeasurementMeasurement

We begin with the measurement of length: its magnitude and its direction.

LengthLength WeightWeight TimeTime

Distance: A Scalar QuantityDistance: A Scalar Quantity

A scalar quantity:

DistanceDistance is the length of the actual path is the length of the actual path taken by an object.taken by an object.

A scalar quantity:

Contains magnitudeonly and consists of a number and a unit.

(20 m, 40 mi/h, 10 gal)

A

Bs = 20 m

DisplacementDisplacementA Vector QuantityA Vector Quantity

A vector quantity:

DisplacementDisplacement is the straightis the straight--line line separation of two points in a specified separation of two points in a specified direction.direction.

A vector quantity:

Contains magnitudeAND direction, a number, unit & angle.

(12 m, 300; 8 km/h, N)

A

BD = 12 m, 20o

Distance and DisplacementDistance and Displacement

Net displacement:Net displacement:

coordinate of coordinate of DisplacementDisplacement is the is the x x or or yy coordinate of coordinate of position. Consider a car that travels 4 position. Consider a car that travels 4 m, E then 6 m, W.m, E then 6 m, W.

Net displacement:Net displacement:

4 m,E4 m,E

6 m,W6 m,W

D

What is the distance What is the distance traveled?traveled?

10 m !!

DD = 2 m, W= 2 m, W

xx = +4= +4xx = = --22

Identifying DirectionIdentifying Direction

A common way of identifying direction A common way of identifying direction A common way of identifying direction A common way of identifying direction is by reference to East, North, West, is by reference to East, North, West, and South. (Locate points below.)and South. (Locate points below.)

N Length = 40 m

40 m, 5040 m, 50oo N of EN of E

EW

S

40 m, 60o N of W

40 m, 60o W of S

40 m, 60o S of E

5050oo60o

60o 60o

Identifying DirectionIdentifying Direction

Write the angles shown below by using Write the angles shown below by using references to east, south, west, north.references to east, south, west, north.

N45o

EW

N

EW

S

45EW

50o

S

Click to see the Answers . . .500 S of E 450 W of N

Vectors and Polar CoordinatesVectors and Polar Coordinates

Polar coordinates (Polar coordinates (R,R,) are an excellent ) are an excellent way to express vectors. Consider the way to express vectors. Consider the vector vector 40 m, 5040 m, 500 0 N of EN of E,, for example.for example.

90o 90o

0o

180o

270o

90

0o

180o

270o

90RR

RR is the is the magnitudemagnitude and and is the is the directiondirection..

40 m40 m

5050oo

Vectors and Polar CoordinatesVectors and Polar Coordinates

(R,(R,) = 40 m, 50) = 40 m, 50oo90o

120o

Polar coordinates (Polar coordinates (R,R,) are given for each ) are given for each of four possible quadrants:of four possible quadrants:

(R,(R,) = 40 m, 120) = 40 m, 120oo

(R,(R,) = 40 m, 210) = 40 m, 210oo

(R,(R,) = 40 m, 300) = 40 m, 300oo

5050oo60o

60o 60o

0o180o

270o

120o

210o

3000

Rectangular CoordinatesRectangular Coordinates

y

(+3, +2)(+3, +2)((--2, +3)2, +3)

Reference is made to Reference is made to xx and and yy axes, with axes, with ++

and and -- numbers to numbers to indicate position in indicate position in space.space.++

++

Right, up = (+,+)

Left, down = (-,-)

(x,y) = (?, ?)

x

(+4, (+4, --3)3)((--1, 1, --3)3)

space.space.++

----

Trigonometry ReviewTrigonometry Review

Application of Trigonometry to Vectors Application of Trigonometry to Vectors

R

y = R sin sin yR

=TrigonometryTrigonometry

y

x

R

x = R cos x = R cos cosx

R =

tanyx

= R2 = x2 + y2

Example 1:Example 1: Find the height of a building Find the height of a building if it casts a shadow if it casts a shadow 90 m90 m long and the long and the indicated angle is indicated angle is 3030oo..

The height h is opposite 300 and the known adjacent side is 90 m.

opp h

90 m

300h

h = (90 m) tan 30o

h = 57.7 m

0tan 3090 m

opp hadj= =

Finding Components of VectorsFinding Components of Vectors

A component is the effect of a vector along other directions. The x and y components of the vector (R,) are illustrated below.

x

yR

x = R cos

y = R sin

Finding components:

Polar to Rectangular Conversions

Example 2:Example 2: A person walks A person walks 400 m400 m in a in a direction of direction of 3030oo N of EN of E. How far is the . How far is the displacement east and how far north?displacement east and how far north?

yR

y = ?

400 m

30

NN

x

x = ?

30E

The y-component (N) is OPP:

The x-component (E) is ADJ: x = R cos

y = R sin

E

Example 2 (Cont.):Example 2 (Cont.): A A 400400--mm walk in a walk in a direction of direction of 3030oo N of EN of E. How far is the . How far is the displacement east and how far north?displacement east and how far north?

y = ?400 m

30

N Note:Note: xx is the side is the side adjacentadjacent to angle to angle 303000

x = R cos

x = (400 m) cos 30o

= +346 m, E

x = ?

y = ?30

E ADJADJ = HYP x = HYP x CosCos 303000

The xThe x--component is:component is:

RRxx = = +346 m+346 m


y = ?400 m

30

N Note:Note: yy is the side is the side oppositeopposite to angle to angle 303000

y = R sin

y = (400 m) sin 30o

= + 200 m, N

x = ?

y = ?30

E OPPOPP = HYP x = HYP x SinSin 303000

The yThe y--component is:component is:

RRyy = = +200 m+200 m


Ry = +200 m

400 m

30

N

The xThe x-- and yand y--components are components are eacheach + in the + in the

Rx = +346 m

+200 m30E

eacheach + in the + in the first quadrantfirst quadrant

Solution: The person is displaced 346 m east and 200 m north of the original position.

Signs for Rectangular CoordinatesSigns for Rectangular Coordinates

First Quadrant:

R is positive (+)

0o > < 90o+

90o

R 0 > < 90

x = +; y = +

x = R cos

y = R sin

+

+

0o


Second Quadrant:

R is positive (+)

90o > < 180o+R

180o

90o

x = - ; y = +

x = R cos

y = R sin

180o


Third Quadrant:

R is positive (+)

180o > < 270o180o

x = - y = -

x = R cos

y = R sin

-R

180o

270o


Fourth Quadrant:

R is positive (+)

270o > < 360o360o+

x = + y = -

x = R cos

y = R sin

360o+

R

270o

Resultant of Perpendicular VectorsResultant of Perpendicular VectorsFinding resultant of two perpendicular vectors is like changing from rectangular to polar coord.

2 2R x y= +y

R

R is always positive; is from + x axis

R x y= +

tanyx

=xy

Example 3:Example 3: A A 3030--lblb southward force southward force and a and a 4040--lblb eastward force act on a eastward force act on a donkey at the same time. What is the donkey at the same time. What is the NET or resultant force on the donkey?NET or resultant force on the donkey?

40 lb

Draw a rough sketch.Draw a rough sketch. Choose rough scale:Choose rough scale:

Ex: 1 cm = 10 lbNote: Force has direction just like length

30 lb

40 lb

4 cm = 40 lb

3 cm = 30 lb

40 lb

30 lb

Note: Force has direction just like length does. We can treat force vectors just as we have length vectors to find the resultant force. The procedure is the same!

Finding Resultant: (Cont.)Finding Resultant: (Cont.)

40 lb 40 lb

Finding (Finding (R,R,) from given () from given (x,yx,y) = (+40, ) = (+40, --30)30)

Ry

Rx

30 lb 30 lbR

R = x2 + y2 R = (40)2 + (30)2 = 50 lb

tan = -3040

= -36.9o = 323.1o

Four Quadrants: (Cont.)Four Quadrants: (Cont.)

40 lb

30 lbR

Ry

Rx

40 lb

30 lb

R

Ry

Rx

R = 50 lb

40 lb

30 lbR

Ry

Rx40 lb

30 lb R

Ry

Rx

40 lb

= 36.9o; = 36.9o; 143.1o; 216.9o; 323.1o

R = 50 lb

Unit vector notation (Unit vector notation (i,j,ki,j,k))

x

y Consider 3D axes (x, y, z)

Define unit vectors, i, j, kij

k Examples of Use:

z

k Examples of Use:

40 m, E = 40 i 40 m, W = -40 i

30 m, N = 30 j 30 m, S = -30 j

20 m, out = 20 k 20 m, in = -20 k

Example 4:Example 4: A woman walks A woman walks 30 m, W30 m, W; ; then then 40 m, N40 m, N. Write her displacement. Write her displacementinin i,ji,j notation and innotation and in R,R, notation.notation.

+40 m R R = Rxi + Ry j

In i,j notation, we have:

-30 m

R = -30 i + 40 j

Rx = - 30 m Ry = + 40 m

Displacement is 30 m west and 40 m north of the starting position.

Example 4 (Cont.):Example 4 (Cont.): Next we find her Next we find her displacementdisplacement inin R,R, notation.notation.

--30 m30 m

+40 +40 mm

R

040tan ; = 59.130

+=

= 180= 18000 59.159.100--30 m30 m

= 126.9o

(R,) = (50 m, 126.9o)

2 2( 30) (40)R = + R = 50 m

Example 6:Example 6: Town A is 35 km south and 46 km Town A is 35 km south and 46 km west of Town B. Find length and direction of west of Town B. Find length and direction of highway between towns.highway between towns.

BB2 2(46 km) (35 km)R = +

46 km46 km

35 35 kmkm

=?=?RR = = --46 46 ii 35 35 jj

R = 57.8 km

46 kmtan

35 km =

= 52.70 S. of W.

kmkmR = ?R = ?

AA

= 232.70 = 180= 18000 + 52.7+ 52.700

Example 7. Example 7. Find the components of the 240Find the components of the 240--N N force exerted by the boy on the girl if his arm force exerted by the boy on the girl if his arm makes an angle of 28makes an angle of 2800 with the ground.with the ground.

282800

FF = 240 N= 240 N

FFFFyy

FFxx

FFyy

FFxx

FFxx = = --|(240 N) cos 28|(240 N) cos 2800|| = = --212 N212 N

FFyy = +|(240 N) sin 28= +|(240 N) sin 2800|| = = +113 N+113 N

Or in Or in i,ji,j notation:notation:

F F = = --(212 N)(212 N)ii + (113 N)+ (113 N)jj

Example 8. Example 8. Find the components of a Find the components of a 300300--N N force acting along the handle of a lawnforce acting along the handle of a lawn--mower. The angle with the ground is mower. The angle with the ground is 323200..

323200

FF = 300 N= 300 N

FF

FFxx

FF

3232oo

3232oo3232

FF FFyyFFyy

FFxx = = --|(300 N) cos 32|(300 N) cos 3200|| = = --254 N254 N

FFyy = = --|(300 N) sin 32|(300 N) sin 3200|| = = --159 N159 N

3232oo

Or in Or in i,ji,j notation:notation:

F F = = --(254 N)(254 N)ii -- (159 N)(159 N)jj

Component MethodComponent Method

1. Start at origin. Draw each vector to scale 1. Start at origin. Draw each vector to scale with tip of 1st to tail of 2nd, tip of 2nd to with tip of 1st to tail of 2nd, tip of 2nd to tail 3rd, and so on for others.tail 3rd, and so on for others.

2. Draw resultant from origin to tip of last 2. Draw resultant from origin to tip of last 2. Draw resultant from origin to tip of last 2. Draw resultant from origin to tip of last vector, noting the quadrant of the resultant.vector, noting the quadrant of the resultant.

3. Write each vector in 3. Write each vector in i,ji,j notation.notation.

4. Add vectors algebraically to get resultant in 4. Add vectors algebraically to get resultant in i,ji,j notation. Then convert to (notation. Then convert to (R,R,).).

Example 9.Example 9. A boat moves A boat moves 2.0 km2.0 km east then east then 4.0 km4.0 km north, then north, then 3.0 km3.0 km west, and finally west, and finally 2.0 km2.0 km south. Find resultant displacement.south. Find resultant displacement.

EE

NN1. Start at origin. 1. Start at origin. Draw each vector to Draw each vector to scale with tip of 1st scale with tip of 1st to tail of 2nd, tip of to tail of 2nd, tip of 2nd to tail 3rd, and 2nd to tail 3rd, and

4 km, N4 km, NBB

3 km, W3 km, WCC2 km, S2 km, S

DD

EE2nd to tail 3rd, and 2nd to tail 3rd, and so on for others.so on for others.

2. Draw resultant from origin to tip of last 2. Draw resultant from origin to tip of last vector, noting the quadrant of the resultant.vector, noting the quadrant of the resultant.

Note: The scale is approximate, but it is still Note: The scale is approximate, but it is still clear that the resultant is in the fourth quadrant.clear that the resultant is in the fourth quadrant.

2 km, E2 km, EAA

Example 9 (Cont.)Example 9 (Cont.) Find resultant displacement.Find resultant displacement.

3.3. Write each vector Write each vector inin i,ji,j notation:notation:

A = +2 A = +2 ii

B = + 4 B = + 4 jj

C = C = --3 3 ii

EE

NN

2 km, E2 km, EAA

4 km, N4 km, NBB


DD

C = C = --3 3 ii

D = D = -- 2 2 jj 4.4. Add vectors A,B,C,D Add vectors A,B,C,D algebraically to get algebraically to get resultant inresultant in i,ji,j notation. notation. RR == --1 1 i i + 2 + 2 jj

1 km, west and 2 km north of origin..

2 km, E2 km, E

5. 5. Convert to Convert to R,R, notation notation See next page.See next page.

Example 9 (Cont.)Example 9 (Cont.) Find resultant displacement.Find resultant displacement.

EE

NN

2 km, E2 km, EAA

4 km, N4 km, NBB


DDResultant Sum is:Resultant Sum is:

RR = = --1 1 ii + 2 + 2 jj

Now, We Find Now, We Find R, R, 2 2( 1) (2) 5R = + = 2 km, E2 km, E

Ry= +2 km

Rx = -1 km

RR

( 1) (2) 5R = + =

R = 2.24 km

2 kmtan

1 km +=

= 63.40 N or W

Reminder of Significant Units:Reminder of Significant Units:

EE

NN

2 km2 kmAA

4 km4 kmBB

3 km3 kmCC2 km2 km

DDFor convenience, we For convenience, we follow the practice of follow the practice of assuming three (3) assuming three (3) significant figures for significant figures for all data in problems.all data in problems. 2 km2 km

In the previous example, we assume that the In the previous example, we assume that the distances are 2.00 km, 4.00 km, and 3.00 km.distances are 2.00 km, 4.00 km, and 3.00 km.

Thus, the answer must be reported as:Thus, the answer must be reported as:

R = 2.24 km, 63.40 N of W

Significant Digits for AnglesSignificant Digits for Angles

40 lb

30 lbR

Ry

Rx

Since a Since a tenthtenth of a of a degreedegree can often be can often be significant, sometimes a significant, sometimes a fourth digit is needed.fourth digit is needed.

Rule: Write angles to 40 lb

30 lbR

Ry

Rx

40 lb

= 36.9o; 323.1o

Rule: Write angles to the nearest tenth of a degree. See the two examples below:

Example 10:Example 10: Find R,Find R, for the three vector for the three vector displacements below: displacements below:

A = 5 mA = 5 m B = 2.1 mB = 2.1 m

202000BB

C = C = 0.5 m0.5 mRR

A = 5 m, 0A = 5 m, 000

B = 2.1 m, 20B = 2.1 m, 2000

C = 0.5 m, 90C = 0.5 m, 9000

1. First draw vectors A, B, and C to approximate 1. First draw vectors A, B, and C to approximate scale and indicate angles. (Rough drawing)scale and indicate angles. (Rough drawing)

2. Draw resultant from origin to tip of last vector; 2. Draw resultant from origin to tip of last vector; noting the quadrant of the resultant. noting the quadrant of the resultant. (R,(R,))

3. Write each vector in 3. Write each vector in i,ji,j notation. (Continued ...)notation. (Continued ...)

Example 10:Example 10: Find Find R,R, for the three vector for the three vector displacements below: (A table may help.)displacements below: (A table may help.)

A = 5 mA = 5 m B = 2.1 mB = 2.1 m

202000BB

C = C = 0.5 m0.5 mRR

For i,j notation find x,y compo-nents of each vector A, B, C.

VectorVector XX--component (component (ii)) YY--component (component (jj))A=5 mA=5 m 0000 + 5 m+ 5 m 00

B=2.1mB=2.1m 202000 +(2.1 m) cos 20+(2.1 m) cos 2000 +(2.1 m) sin 20+(2.1 m) sin 2000

C=.5 mC=.5 m 909000 00 + 0.5 m+ 0.5 m

RRxx = A= Axx+B+Bxx+C+Cxx RRyy = A= Ayy+B+Byy+C+Cyy

Example 10 (Cont.):Example 10 (Cont.): Find Find i,ji,j for three for three vectors: vectors: A A = 5 m,0= 5 m,000; ; BB = 2.1 m, 20= 2.1 m, 2000; ; CC = 0.5 m, = 0.5 m, 909000..

XX--component (component (ii)) YY--component (component (jj))

AAxx = + 5.00 m= + 5.00 m AAyy = 0= 0

BBxx = +1.97 m= +1.97 m BByy = +0.718 m= +0.718 m

CC = 0= 0 CC = + 0.50 m= + 0.50 mCCxx = 0= 0 CCyy = + 0.50 m= + 0.50 m

AA = 5.00 = 5.00 i i + 0 + 0 jj

BB = 1.97 = 1.97 ii + 0.718 + 0.718 jj

CC = 0 = 0 i i + 0.50+ 0.50 jj

4. Add vectors to 4. Add vectors to get resultant get resultant R R in in i,ji,j notation.notation.

RR == 6.97 6.97 i i + 1.22 + 1.22 jj

Example 10 (Cont.): Example 10 (Cont.): Find Find i,ji,j for three vectors: for three vectors: A A = 5 m,0= 5 m,000; ; BB = 2.1 m, 20= 2.1 m, 2000; ; CC = 0.5 m, 90= 0.5 m, 9000..

2 2(6.97 m) (1.22 m)R = +

RR = = 6.97 6.97 i i + 1.22 + 1.22 jj

5. Determine R,5. Determine R, from x,y:from x,y:RR

RRy y

1.22 m1.22 m

Diagram for Diagram for finding R,finding R,::

2 2(6.97 m) (1.22 m)R = +

R = 7.08 m

1.22 mtan

6.97 m = = 9.930 N. of E.

RRxx= 6.97 m= 6.97 m

y y

1.22 m1.22 m

Example 11:Example 11: A bike travels A bike travels 20 m, E20 m, E then then 40 m40 mat at 6060oo N of WN of W, and finally , and finally 30 m30 m at at 210210oo. What . What is the resultant displacement graphically?is the resultant displacement graphically?

30o

Graphically, we use ruler and protractor to draw components, then measure the

B = 40 m

C = 30 m

60o

R

then measure the Resultant R,

A = 20 m, E

R = (32.6 m, 143.0o)Let 1 cm = 10 m

A Graphical Understanding of the Components A Graphical Understanding of the Components and of the Resultant is given below:and of the Resultant is given below:

30o

R

Note: Rx = Ax + Bx + Cx

BC Ry = Ay + By + Cy0By

Cy

60o

R

Ax

Bx

Rx

A

Cx

Ry

Example 11 (Cont.)Example 11 (Cont.) Using the Component Using the Component Method to solve for the ResultantMethod to solve for the Resultant..

60

30o

R

B

A

C

Ry

ByCy

Write each vector in i,j notation.

Ax = 20 m, Ay = 0

A = 20 i60 A

xBx

Rx

A

Cx

Bx = -40 cos 60o = -20 m

By = 40 sin 60o = +34.6 m

Cx = -30 cos 30o = -26 m

Cy = -30 sin 60o = -15 m

B = -20 i + 34.6 j

C = -26 i - 15 j

A = 20 i

Example 11 (Cont.)Example 11 (Cont.) The Component MethodThe Component Method

60

30o

R

B

A

C

Ry

ByCy

Add algebraically:Add algebraically:

A = 20 i

B = -20 i + 34.6 j

C = -26 i - 15 jA

xBx

Rx

Cx

C = -26 i - 15 j

R = -26 i + 19.6 j

R

-26

+19.6 R = (-26)2 + (19.6)2 = 32.6 m

tan = 19.6-26

= 143o

Example 11 (Cont.)Example 11 (Cont.) Find the Resultant.Find the Resultant.

60

30o

R

B

A

C

Ry

ByCy RR = = --26 i + 19.6 j26 i + 19.6 j

R+19.6

A

xBx

Rx

Cx

-26

The Resultant Displacement of the bike is best The Resultant Displacement of the bike is best given by its polar coordinates given by its polar coordinates RR and and ..

R = 32.6 m; = 1430

Example 12.Example 12. Find A + B + C for Vectors Find A + B + C for Vectors Shown below.Shown below.

A = 5 m, 900

B = 12 m, 00

C = 20 m, -350

A

B

RR

C350CCxx

CCyy

AAxx = 0; = 0; AAyy = +5 m= +5 m

BBxx = +12 m; = +12 m; BByy = 0= 0

CCxx = (20 m) cos 35= (20 m) cos 3500

CCyy = = --(20 m) sin (20 m) sin --353500

AA = 0 = 0 i i + 5.00 + 5.00 jj

BB = 12 = 12 ii + 0 + 0 jj

CC = 16.4 = 16.4 i i 11.511.5 jj

RR == 28.4 28.4 i i -- 6.47 6.47 jj

Example 12 (Continued).Example 12 (Continued). Find A + B + C Find A + B + C

A

B

C

350

RR

RR

Rx = 28.4 m

Ry = -6.47 m

2 2(28.4 m) (6.47 m)R = + R = 29.1 m6.47 m

tan28.4 m

= = 12.80 S. of E.

Vector DifferenceVector Difference

For vectors, signs are indicators of direction. For vectors, signs are indicators of direction. Thus, when a vector is subtracted, the sign Thus, when a vector is subtracted, the sign (direction) must be changed before adding.(direction) must be changed before adding.

First Consider A + BA + B Graphically:First Consider A + BA + B Graphically:

B

A

BR = A + B

RR

AB



Now A B: First change sign (direction) Now A B: First change sign (direction) of B, then add the negative vector.

B

A

B --B

A

--BRR

A

Comparison of addition and subtraction of B

Addition and SubtractionAddition and Subtraction

Subtraction results in a significant difference Subtraction results in a significant difference both in the both in the magnitudemagnitude and the and the directiondirection of of the resultant vector. the resultant vector. |(A |(A B)| = |A| B)| = |A| -- |B||B|

Comparison of addition and subtraction of B

B

A

BR = A + B

RR

AB --BRR

A

R = A - B

Example 13.Example 13. Given Given A = 2.4 km, NA = 2.4 km, N and and B = 7.8 km, NB = 7.8 km, N: find : find A A BB and and B B AA..

A A B; B; B B -- AA

A - B

+A

B - A

+B-A

A A 2.43 N2.43 N

B B 7.74 N7.74 N

-B

(2.43 N 7.74 S)

5.31 km, S

+B

(7.74 N 2.43 S)

5.31 km, N

RR RR

Summary for VectorsSummary for Vectors

A A scalar quantityscalar quantity is completely specified is completely specified by its magnitude only. (by its magnitude only. (40 m40 m, , 10 gal10 gal))

A A vector quantityvector quantity is completely specified by is completely specified by its magnitude its magnitude andand direction. (direction. (40 m, 3040 m, 3000))its magnitude its magnitude andand direction. (direction. (40 m, 3040 m, 30 ))

Rx

RyR

Components of R:Components of R:

RRxx = R = R coscos RRyy = R = R sin sin

Summary Continued:Summary Continued:

Resultant of Vectors:Resultant of Vectors:

Finding the Finding the resultantresultant of two perpendicular of two perpendicular vectors is like converting from polar (R, vectors is like converting from polar (R, ) ) to the rectangular (Rto the rectangular (Rxx, R, Ryy) coordinates.) coordinates.

Rx

RyR

Resultant of Vectors:Resultant of Vectors:

2 2R x y= +

tanyx

=

Component Method for VectorsComponent Method for Vectors

Start at origin and draw each vector in Start at origin and draw each vector in succession forming a labeled polygon.succession forming a labeled polygon.

Draw resultant from origin to tip of last Draw resultant from origin to tip of last vector, noting the quadrant of resultant.vector, noting the quadrant of resultant.vector, noting the quadrant of resultant.vector, noting the quadrant of resultant.

Write each vector in Write each vector in i,ji,j notation (notation (RRxx,R,Ryy).).

Add vectors algebraically to get resultant Add vectors algebraically to get resultant in in i,ji,j notation. Then convert to (notation. Then convert to (RR,,).).



Now A B: First change sign (direction) Now A B: First change sign (direction) of B, then add the negative vector.

B

A

B --B

A

--BRR

A

Conclusion of Chapter 3B Conclusion of Chapter 3B -- VectorsVectors

Chapter3B Vectors

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Transcript of Chapter3B Vectors