Chapter 3B - - Vectors - St. Charles Preparatory School Links... · Chapter 3B - - Vectors ......

Chapter 3B Chapter 3B -- VectorsVectorsA PowerPoint Presentation by

Paul E. Tippens, Professor of Physics

Southern 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

Surveyors use accurate measures of magnitudes and directions to create

scaled maps of large regions.

VectorsVectors

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

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

• Define and give examples of scalar and vector quantities.

• Determine the components of a given vector.

• Find the resultant of two or more vectors.

•• 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 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.

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

2fv v

x t

Solve for vo

0

2fv t xv

t

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:

(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 nNF = 6.94 x 10-9 N = 6.94 nN

Expectations (Continued)Expectations (Continued)

•• 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

Expectations (Continued)Expectations (Continued)•• You must have mastered rightYou must have mastered right--triangle triangle

trigonometry. trigonometry.

y

x

R

y = R sin y = R sin

x = R cos x = R cos

sin yR

cos xR

tan yx

R2 = x2 + y2R2 = x2 + y2

Mathematics ReviewMathematics Review

Select Chap. 2 from the On-Line Learning Center in Tippens—Student Edition

Select Chap. 2 from the On-Line Learning Center in Tippens—Student Edition

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 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.

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:

Contains magnitude only and consists of a number and a unit.

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

A

B

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

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

s = 20 m

DisplacementDisplacement——A Vector QuantityA Vector Quantity

A vector quantity:

Contains magnitude AND direction, a number, unit & angle.

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

A

BD = 12 m, 20o

• Displacement is the straight-line separation of two points in a specified direction.

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

Distance and DisplacementDistance and Displacement

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

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

•• 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.

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

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

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.)

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

EW

S

N

40 m, 60o N of W

40 m, 60o W of S

40 m, 60o S of E

Length = 40 m

5050oo60o

60o 60o

Identifying DirectionIdentifying Direction

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

EW

S

N45o

EW

N

50o

S

Click to see the Answers . . .Click to see the Answers . . .500 S of E500 S of E 450 W of N450 W of N

Vectors and Polar CoordinatesVectors and Polar CoordinatesPolar coordinates (R,) are an excellent way to express vectors. Consider the vector 40 m, 500 N of E, for example.

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.

0o

180o

270o

90o

0o

180o

270o

90o

RR

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, 50oo

(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

90o

120o

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

210o

3000

Rectangular CoordinatesRectangular Coordinates

Right, up = (+,+)

Left, down = (-,-)

(x,y) = (?, ?)

x

y

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

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

++

----

Trigonometry ReviewTrigonometry Review•• Application of Trigonometry to Vectors Application of Trigonometry to Vectors

y

x

R

y = R sin y = R sin

x = R cos x = R cos

sin yR

cos xR

tan yx

R2 = x2 + y2R2 = x2 + y2

TrigonometryTrigonometry

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..

90 m

300

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

h

h = (90 m) tan 30o

h = 57.7 mh = 57.7 m

0tan 3090 m

opp hadj

Finding Components of VectorsFinding Components of VectorsA 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?

x

yR

x = ?

y = ?400 m

E

N

The y-component (N) is OPP:

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

y = R sin

E

N

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?

x = R cos

x = (400 m) cos 30o

= +346 m, E

x = ?

y = ?400 m

E

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

ADJADJ = HYP x = HYP x CosCos 303000

The xThe x--component is:component is:

RR xx = = +346 m+346 m


y = R sin

y = (400 m) sin 30o

= + 200 m, N

x = ?

y = ?400 m

E

N

OPPOPP = HYP x = HYP x SinSin 303000

The yThe y--component is:component is:

RR yy = = +200 m+200 m

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


Rx = +346 m

Ry = +200 m

400 m

E

NThe xThe x-- and yand y--

components are components are 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

x = +; y = +

x = R cos

y = R sin

+

+0o

90o

R


Second Quadrant:

R is positive (+)

90o > < 180o

x = - ; y = +

x = R cos

y = R sin

+R

180o

90o


Third Quadrant:

R is positive (+)

180o > < 270o

x = - y = -

x = R cos

y = R sin

-R

180o

270o


Fourth Quadrant:

R is positive (+)

270o > < 360o

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.

R is always positive;

is from + x axis

2 2R x y

tan yx

x

yR

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?

30 lb

40 lb

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

Ex: 1 cm = 10 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!

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

30 lb

40 lb

30 lb

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

R

Ry

Rx

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

tan = -30

40

= -36.9o

= 323.1o

= 323.1o

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

40 lb

30 lbR

Ry

Rx40 lb

30 lb R

Ry

Rx

40 lb

30 lbR

Ry

Rx

40 lb

30 lb

R

Ry

Rx

= 36.9o;

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

= 36.9o;

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

R = 50 lb

R = 50 lb

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

x

z

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

Define unit vectors, i, j, kij

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 displacement inin i,ji,j notation and innotation and in R,R,

notation.notation.

-30 m

+40 m R

R = Rx i + Ry j

R = -30 i + 40 jR = -30 i + 40 j

Rx = - 30 m Ry = + 40 m

In i,j notation, we have:

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

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

= 126.9o

= 126.9o

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

040tan ; = 59.130

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

= 180= 18000 –– 59.159.100

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

R = 57.8 kmR = 57.8 km

46 kmtan35 km

= 52.70 S. of W.

= 52.70 S. of W.

46 km46 km

35 35 kmkm

R = ?R = ?

AA

RR = = --46 46 ii –– 35 35 jj

= 232.70

= 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

FF FF yyFF xx

FFyy

FFxx = = --|(240 N) |(240 N) coscos 282800|| = = --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 (212 N)N)ii + (113 + (113 N)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 FF yy

FF xx

FFyy

FFxx = = --|(300 N) |(300 N) coscos 323200|| = = --254 N254 N

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

3232oo

3232oo

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

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

Component MethodComponent Method1. 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 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 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

4 km, N4 km, NBB

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

DD

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

EE

NN

2 km, E2 km, EAA

4 km, N4 km, NBB


DD

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

Ry = +2 km

Rx = -1 km

RR

Now, We Find Now, We Find R, R, 2 2( 1) (2) 5R

R = 2.24 km

2 kmtan1 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.

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 WR = 2.24 km, 63.40 N of W

Significant Digits for AnglesSignificant Digits for Angles

40 lb

30 lbR

Ry

Rx

40 lb

30 lbR

Ry

Rx

= 36.9o; 323.1o

= 36.9o; 323.1o

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 the nearest tenth of a degree. See the two examples below:

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.)

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) +(2.1 m) coscos 202000 +(2.1 m) sin 20+(2.1 m) sin 2000

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

RR xx = = AA xx +B+B xx +C+C xx RR yy = A= A yy +B+B yy +C+C yy

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 components of each vector A, B, C.

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))

AA xx = + 5.00 m= + 5.00 m AA yy = 0= 0

BB xx = +1.97 m= +1.97 m BB yy = +0.718 m= +0.718 m

CC xx = 0= 0 CC yy = + 0.50 m= + 0.50 m

AA = 5.00 = 5.00 i i + 0 + 0 jjBB = 1.97 = 1.97 ii + 0.718 + 0.718 jjCC = 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

R = 7.08 mR = 7.08 m

1.22 mtan6.97 m

= 9.930 N. of E. = 9.930 N. of E.

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

5. Determine R,5. Determine R,

from x,y:from x,y:

RRxx = 6.97 m= 6.97 m

RR

RRyy

1.22 m1.22 m

Diagram for Diagram for finding R,finding R,

Example 11:Example 11: A bike travels A bike travels 20 m, E20 m, E then then 40 m40 m at 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?

60o

30o

R

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

A = 20 m, E

B = 40 m

C = 30 m

R = (32.6 m, 143.0o)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:

60o

30o

R

Note: Rx = Ax + Bx + Cx

Ax

B

Bx

Rx

A

C

Cx

Ry = Ay + By + Cy

0

Ry

ByCy

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

Ax

B

Bx

Rx

A

C

Cx

Ry

ByCy

Write each vector in i,j notation.

Ax = 20 m, Ay = 0

Bx = -40 cos 60o = -20 m

By = 40 sin 60o = +34.6 m

Cx = -30 cos 30o = -26 mCy = -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

Ax

B

Bx

Rx

A

C

Cx

Ry

ByCy

Add algebraically:Add algebraically:A = 20 i

B = -20 i + 34.6 j

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

= 143o

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

60

30o

R

Ax

B

Bx

Rx

A

C

Cx

Ry

ByCy

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

R

-26

+19.6

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;

= 1430R = 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

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

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

CCxx = (20 m) = (20 m) coscos 353500

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

AA = 0 = 0 i i + 5.00 + 5.00 jjBB = 12 = 12 ii + 0 + 0 jjCC = 16.4 = 16.4 i i –– 11.511.5 jj

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

C

CCxx

CCyy

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

A

B

C

RR RR

Rx = 28.4 m

Ry = -6.47 m

2 2(28.4 m) (6.47 m)R R = 29.1 mR = 29.1 m

6.47 mtan28.4 m

= 12.80 S. of E. = 12.80 S. of E.

Vector DifferenceVector DifferenceFor 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:

B

A

BR = A + B

RR

AB


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

B

A

B

Addition and SubtractionAddition and Subtraction

R = A + B

RR

AB --BRR’’

AR’ = A - B

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|

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 2.43 N2.43 N

B B 7.74 N7.74 N

A A –– B; B; B B -- AA

A - B

+A

-B

(2.43 N – 7.74 S)

5.31 km, S

B - A

+B-A

(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))

Rx

RyR

Components of R:Components of R:

RR xx = R = R coscos

RR yy = R = R sin sin

Summary Continued:Summary Continued:

Rx

RyR

Resultant of Vectors:Resultant of Vectors:2 2R x y

tan yx

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 (R xx , , RR yy ) coordinates.) coordinates.

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.

Write each vector in Write each vector in i,ji,j notation (notation (RR xx ,R,R yy ).).

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) of B, then add the negative vector.B

A

B --B

A--BRR’’

A

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

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