Chapter 1 – Math Chapter 1 – Math ReviewReview

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

EE

NN1. Start at origin. 1. Start at origin. Draw each vector Draw each vector to scale with tip of to scale with tip of 1st to tail of 2nd, 1st to tail of 2nd, tip of 2nd to tail tip of 2nd to tail 3rd, and so on for 3rd, and so on for others.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 vector, noting the quadrant of the resultant.resultant.Note: The scale is approximate, but it is Note: The scale is approximate, but it is still clear that the resultant is in the fourth still clear that the resultant is in the fourth quadrant.quadrant.

2 km, 2 km, EE

AA

4 km, N4 km, NBB

3 km, W3 km, WCC2 km, 2 km,

SS

DD

Example 9 (Cont.)Example 9 (Cont.) Find resultant Find resultant displacement.displacement.3.3. Write each Write each

vector invector in i,ji,j notation:notation:A = +2 A = +2 ii

B = + 4 B = + 4 jjC = -3 C = -3 ii

D = - 2 D = - 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, 2 km, EE

AA

4 km, N4 km, NBB

3 km, W3 km, WCC2 km, 2 km,

SS

DD

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

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

EE

NN

2 km, 2 km, EE

AA

4 km, N4 km, NBB

3 km, W3 km, WCC2 km, 2 km,

SS

DDResultant Sum Resultant Sum is: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 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, For convenience, we follow the we follow the practice of practice of assuming three assuming three (3) significant (3) significant figures for all data figures for all data in problems.in problems.In the previous example, we assume In the previous example, we assume that the distances are 2.00 km, 4.00 km, that the distances are 2.00 km, 4.00 km, and 3.00 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 Significant Digits for AnglesAngles

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, significant, sometimes a fourth sometimes a fourth digit is needed.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 for the three vector displacements below: vector displacements below:

A = 5 mA = 5 m B = 2.1 B = 2.1 mm

202000BB

C = C = 0.5 m0.5 mRR

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

B = 2.1 m, B = 2.1 m, 202000

C = 0.5 m, C = 0.5 m, 909000

1. First draw vectors A, B, and C to 1. First draw vectors A, B, and C to approximate scale and indicate angles. approximate scale and indicate angles. (Rough drawing)(Rough drawing)2. Draw resultant from origin to tip of last 2. Draw resultant from origin to tip of last vector; noting the quadrant of the vector; noting the quadrant of the resultant. resultant. (R,(R,))

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

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

VectorVector

X-component X-component ((ii))

Y-component Y-component ((jj))

A=5 A=5 mm

0000 + 5 m+ 5 m 00

B=2.1B=2.1mm

202000

+(2.1 m) cos +(2.1 m) cos 202000

+(2.1 m) sin +(2.1 m) sin 202000

C=.5 C=.5 mm

909000

00 + 0.5 m+ 0.5 m

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

A = 5 mA = 5 m B = 2.1 B = 2.1 mm

202000BB

C = C = 0.5 m0.5 mRR

For i,j notation find x,y compo-nents 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 = 0.5 m, 90m, 9000..

X-component X-component ((ii))

Y-component Y-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

CCxx = 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 4. Add vectors to get to get resultant 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 for three vectors: vectors: A A = 5 m,0= 5 m,000; ; BB = 2.1 m, 20 = 2.1 m, 2000; ; CC = 0.5 = 0.5

m, 90m, 9000..

2 2(6.97 m) (1.22 m)R

R = 7.08 mR = 7.08 m

1.22 mtan

6.97 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 from x,y:x,y:

RRxx= 6.97 = 6.97 mm

RR

RRy y 1.22 1.22 mm

Diagram Diagram for finding for finding R,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 W N of W, and finally , and finally 30 m30 m at at 210210oo. What is the resultant . What is the resultant displacement graphically?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 A Graphical Understanding of the Components and of the Resultant is Components and of the Resultant is given below:given below:

60o

30o

R

Note: Rx = Ax + Bx + Cx

Ax

B

Bx

Rx

A

C

Cx

Ry = Ay + By + Cy

0

Ry

By

Cy

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

A

x

B

Bx

Rx

A

C

Cx

Ry

By

Cy

Write each vector in i,j notation.

Ax = 20 m, Ay = 0

Bx = -40 cos 60o = -20 mBy = 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 The Component MethodMethod

60

30o

R

A

x

B

Bx

Rx

A

C

Cx

Ry

By

Cy

Add Add algebraically:algebraically:A = 20 i

B = -20 i + 34.6 jC = -26 i - 15 j

R = -26 i + 19.6 j

R

-26

+19.6

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

19.6

-26 = 143o = 143o

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

60

30o

R

A

x

B

Bx

Rx

A

C

Cx

Ry

By

Cy

RR = -26 i + 19.6 = -26 i + 19.6 jj

R

-26

+19.6

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

R = 32.6 m; = 1430

R = 32.6 m; = 1430

Example 12.Example 12. Find A + B + C for Find A + B + C for Vectors Shown below.Vectors 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 = = 00CCxx = (20 m) cos = (20 m) cos

353500

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 – 11.5– 11.5 j j

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 mtan

28.4 m = 12.80 S. of E. = 12.80 S. of E.

Vector DifferenceVector Difference

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

First Consider A + BA + B Graphically:

B

A

BR = A + B

RR

AB

Vector DifferenceVector Difference

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

Now A – B: First change sign (direction) of B, then add the

negative vector.B

A

B --B

A

--BR’R’

A

Comparison of addition and subtraction of B

B

A

B

Addition and SubtractionAddition and Subtraction

R = A + B

RR

AB --BR’R’

AR’ = A - B

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

A A 2.43 2.43

NN

B B 7.74 7.74

NN

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 is completely

specified by its magnitude only. (specified by its magnitude only. (40 40 mm, , 10 gal10 gal)) A A vector quantityvector quantity is completely is completely specified by its magnitude specified by its magnitude andand direction. (direction. (40 m, 3040 m, 3000))

Rx

Ry

R

Components of R:Components of R:

RRxx = R = R coscos

RRyy = R = R sin sin

Summary Continued:Summary Continued:

Rx

Ry

R

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

tany

x

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

Component Method for Component Method for VectorsVectors

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

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

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

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

Vector DifferenceVector Difference

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

Now A – B: First change sign (direction) of B, then add the

negative vector.B

A

B --B

A

--BR’R’

A

Conclusion of Chapter - VectorsConclusion of Chapter - Vectors