Chapter 3B  Vectors. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University. © 2007. Surveyors use accurate measures of magnitudes and directions to create scaled maps of large regions. Vectors.
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A PowerPoint Presentation by
Paul E. Tippens, Professor of Physics
Southern Polytechnic State University
© 2007
Surveyors use accurate measures of magnitudes and directions to create scaled maps of large regions.
Vectorss
1 km
1000 m
3600 s
1 h
40 x  x  = 144 km/h
ExpectationsConvert 40 m/s into kilometers per hour.
(8.77 x 103)2
Gmm’
r2
F =  = 
Expectations (Continued)Evaluate the following:
F = 6.94 x 109 N = 6.94 nN
The meter (m) 1 m = 1 x 100 m
1 Gm = 1 x 109 m 1 nm = 1 x 109 m
1 Mm = 1 x 106 m 1 mm = 1 x 106 m
1 km = 1 x 103 m 1 mm = 1 x 103 m
y
q
x
Expectations (Continued)y = R sin q
x = R cos q
R2 = x2 + y2
If you feel you need to brush up on your mathematics skills, try the tutorial from Chap. 2 on Mathematics. Trig is reviewed along with vectors in this module.
Mathematics ReviewSelect Chap. 2 from the OnLine Learning Center in Tippens—Student Edition
Time
Physics is the Science of MeasurementLength
We begin with the measurement of length: its magnitude and its direction.
A
Distance: A Scalar QuantityA scalar quantity:
Contains magnitude only and consists of a number and a unit.
(20 m, 40 mi/h, 10 gal)
s = 20 m
D = 12 m, 20o
A
q
Displacement—A Vector QuantityA vector quantity:
Contains magnitude AND direction, a number,unit & angle.
(12 m, 300; 8 km/h, N)
4 m,E
6 m,W
Distance and DisplacementNet displacement:
D= 2 m, W
What is the distance traveled?
x= 2
x= +4
10 m !!
60o
50o
W
E
60o
60o
S
Identifying DirectionA common way of identifying direction is by reference to East, North, West, and South. (Locate points below.)
Length = 40 m
40 m, 50o N of E
40 m, 60o N of W
40 m, 60o W of S
40 m, 60o S of E
45o
N
W
E
W
E
50o
S
S
Identifying DirectionWrite the angles shown below by using references to east, south, west, north.
500 S of E
Click to see the Answers . . .
450 W of N
90o
R
180o
180o
50o
q
0o
0o
270o
270o
Vectors and Polar CoordinatesPolar coordinates (R,q) are an excellent way to express vectors. Consider the vector 40 m, 500 N of E, for example.
40 m
R is the magnitude and q is the direction.
180o
0o
60o
50o
60o
60o
3000
210o
270o
120o
Vectors and Polar CoordinatesPolar coordinates (R,q) are given for each of four possible quadrants:
(R,q) = 40 m, 50o
(R,q) = 40 m, 120o
(R,q) = 40 m, 210o
(R,q) = 40 m, 300o
(2, +3)
(+3, +2)
+
+
x

Right, up = (+,+)
Left, down = (,)
(x,y) = (?, ?)

(1, 3)
(+4, 3)
Rectangular CoordinatesReference is made to x and y axes, with + and numbers to indicate position in space.
y
q
x
Trigonometry ReviewTrigonometry
y = R sin q
x = R cos q
R2 = x2 + y2
90 m
Example 1:Find the height of a building if it casts a shadow 90 m long and the indicated angle is 30o.The height h is opposite 300 and the known adjacent side is 90 m.
h
h = (90 m) tan 30o
h = 57.7 m
y
q
x
Finding Components of VectorsA component is the effect of a vector along other directions. The x and y components of the vector (R,q) are illustrated below.
x = R cos q
y = R sin q
Finding components:
Polar to Rectangular Conversions
400 m
y = ?
30o
E
x = ?
R
y
q
x
Example 2:A person walks 400 m in a direction of 30o N of E. How far is the displacement east and how far north?N
E
The xcomponent (E) is ADJ:
x = R cosq
The ycomponent (N) is OPP:
y = R sinq
400 m
y = ?
30o
E
x = ?
The xcomponent is:
Rx = +346 m
Example 2 (Cont.):A 400m walk in a direction of 30o N of E. How far is the displacement east and how far north?Note:x is the side adjacent to angle 300
ADJ = HYP x Cos 300
x = R cosq
x = (400 m)cos30o
= +346 m, E
400 m
y = ?
30o
E
x = ?
The ycomponent is:
Ry = +200 m
Example 2 (Cont.):A 400m walk in a direction of 30o N of E. How far is the displacement east and how far north?Note:y is the side opposite to angle 300
OPP = HYP x Sin 300
y = R sinq
y = (400 m) sin 30o
= + 200 m, N
The x and y components are each + in the first quadrant
400 m
Ry = +200 m
30o
E
Rx = +346 m
Example 2 (Cont.):A 400m walk in a direction of 30o N of E. How far is the displacement east and how far north?Solution: The person is displaced 346 m east and 200 m north of the original position.
y = R sinq
Signs for Rectangular Coordinates90o
First Quadrant:
R is positive (+)
0o > q < 90o
x = +; y = +
R
+
q
0o
+
y = R sinq
Signs for Rectangular Coordinates90o
Second Quadrant:
R is positive (+)
90o > q < 180o
x =  ; y = +
R
+
q
180o
y = R sin q
Signs for Rectangular CoordinatesThird Quadrant:
R is positive (+)
180o > q < 270o
x =  y = 
q
180o

R
270o
y = R sin q
Signs for Rectangular CoordinatesFourth Quadrant:
R is positive (+)
270o > q < 360o
x = + y = 
q
+
360o
R
270o
Finding resultant of two perpendicular vectors is like changing from rectangular to polar coord.
R
y
q
x
R is always positive; q is from + x axis
40 lb
30 lb
30 lb
Example 3:A 30lb southward force and a 40lb eastward force act on a donkey at the same time. What is the NET or resultant force on the donkey?
Draw a rough sketch.
Choose rough scale:
Ex:1 cm = 10 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!
4 cm = 40 lb
3 cm = 30 lb
40 lb
f
40 lb
30 lb
R = (40)2 + (30)2 = 50 lb
R = x2 + y2
30
40
30 lb
tan f =
Finding Resultant: (Cont.)Finding (R,q) from given (x,y) = (+40, 30)
Rx
Ry
R
q = 323.1o
f = 36.9o
30 lb
R
R
R = 50 lb
q
40 lb
Ry
q
Ry
f
f
q
Rx
Rx
40 lb
40 lb
R
30 lb
Rx
Rx
q
40 lb
f
Ry
Ry
R = 50 lb
30 lb
R
Four Quadrants: (Cont.)f = 36.9o; q = 36.9o; 143.1o; 216.9o; 323.1o
j
i
x
k
z
Unit vector notation (i,j,k)Consider 3D axes (x, y, z)
Define unit vectors, i, j, 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
+40 m
f
30 m
Example 4:A woman walks 30 m, W; then 40 m, N. Write her displacementini,jnotation and inR,qnotation.In i,j notation, we have:
R = Rxi + Ry j
Rx =  30 m
Ry = + 40 m
R = 30 i + 40 j
Displacement is 30 m west and 40 m north of the starting position.
+40 m
f
30 m
Example 4 (Cont.):Next we find her displacementinR,qnotation.q= 1800 – 59.10
q = 126.9o
R = 50 m
(R,q) = (50 m, 126.9o)
f=?
B
35 km
R = ?
A
q = 1800 + 52.70
q = 232.70
Example 6:Town A is 35 km south and 46 km west of Town B. Find length and direction of highway between towns.R = 46 i – 35 j
R = 57.8 km
f = 52.70 S. of W.
Fy
F
280
Or in i,j notation:
Fy
F = (212 N)i+ (113 N)j
Fx
Example 7. Find the components of the 240N force exerted by the boy on the girl if his arm makes an angle of 280 with the ground.Fx= (240 N) cos 280= 212 N
Fy= +(240 N) sin 280= +113 N
32o
32o
Fx
320
Fy
Fy
F
Or in i,j notation:
F = (254 N)i (159 N)j
Example 8. Find the components of a 300N force acting along the handle of a lawnmower. The angle with the ground is 320.Fx= (300 N) cos 320= 254 N
Fy= (300 N) sin 320= 159 N
1. Start at origin. Draw each vector to scale with tip of 1st to tail of 2nd, tip of 2nd to tail 3rd, and so on for others.
Component Method2. Draw resultant from origin to tip of last vector, noting the quadrant of the resultant.
3. Write each vector in i,j notation.
4. Add vectors algebraically to get resultant in i,j notation. Then convert to (R,q).
B
3 km, W
4 km, N
C
E
A
D
2 km, E
2 km, S
Example 9.A boat moves 2.0 km east then 4.0 km north, then 3.0 km west, and finally 2.0 km south. Find resultant displacement.1. Start at origin. Draw each vector to scale with tip of 1st to tail of 2nd, tip of 2nd to tail 3rd, and so on for others.
2. Draw resultant from origin to tip of last vector, noting the quadrant of the resultant.
Note: The scale is approximate, but it is still clear that the resultant is in the fourth quadrant.
B
3 km, W
4 km, N
C
E
A
D
2 km, S
2 km, E
5. Convert to R,q notation See next page.
Example 9 (Cont.)Find resultant displacement.3.Write each vector ini,jnotation:
A = +2 i
B = + 4 j
C = 3 i
D =  2 j
4.Add vectors A,B,C,D algebraically to get resultant ini,jnotation.
1 i
+ 2 j
R =
1 km, west and 2 km north of origin.
R = 1 i + 2 j
N
B
3 km, W
4 km, N
C
E
D
A
2 km, S
2 km, E
Ry= +2 km
R
f
Rx = 1 km
Example 9 (Cont.)Find resultant displacement.Now, We Find R,
R = 2.24 km
= 63.40 N or W
For convenience, we follow the practice of assuming three (3) significant figures for all data in problems.
D
3 km
2 km
C
B
4 km
E
A
2 km
Reminder of Significant Units:In the previous example, we assume that the distances are 2.00 km, 4.00 km, and 3.00 km.
Thus, the answer must be reported as:
R = 2.24 km, 63.40 N of W
R
q
40 lb
Ry
f
q
Rx
40 lb
R
30 lb
Rx
Ry
Significant Digits for AnglesSince a tenthof a degree can often be significant, sometimes a fourth digit is needed.
Rule:Write angles to the nearest tenth of a degree. See the two examples below:
q = 36.9o; 323.1o
C = 0.5 m
R
B = 2.1 m, 200
B
q
C = 0.5 m, 900
200
A = 5 m
B = 2.1 m
Example 10: Find R,q for the three vector displacements below:1. First draw vectors A, B, and C to approximate scale and indicate angles. (Rough drawing)
2. Draw resultant from origin to tip of last vector; noting the quadrant of the resultant. (R,q)
3. Write each vector in i,j notation. (Continued ...)
For i,j notation find x,y components of each vector A, B, C.
C = 0.5 m
R
B
q
200
A = 5 m
B = 2.1 m
Example 10: Find R,q for the three vector displacements below: (A table may help.)B = 1.97 i + 0.718 j
C = 0 i + 0.50 j
6.97 i
+ 1.22 j
Example 10 (Cont.): Find i,j for three vectors: A = 5 m,00; B = 2.1 m, 200; C = 0.5 m, 900.4. Add vectors to get resultant R in i,j notation.
R =
Diagram for finding R,q:
R
q
Ry 1.22 m
Rx= 6.97 m
Example 10 (Cont.): Find i,j for three vectors: A = 5 m,00; B = 2.1 m, 200; C = 0.5 m, 900.5. Determine R,q from x,y:
R = 7.08 m
q = 9.930 N. of E.
Example 11:A bike travels 20 m, E then 40 m at 60o N of W, and finally 30 m at 210o. What is the resultant displacement graphically?
C = 30 m
Graphically, we use ruler and protractor to draw components, then measure the Resultant R,q
B = 40 m
30o
R
60o
f
A = 20 m, E
R = (32.6 m, 143.0o)
Let 1 cm = 10 m
30o
R
Ry
60o
f
0
q
Rx
Ax
Cx
Bx
A Graphical Understanding of the Components and of the Resultant is given below:Note: Rx = Ax + Bx + Cx
By
B
Ry = Ay + By + Cy
C
A
By
30o
B
C
R
Ry
A
60
f
q
Rx
Ax
Cx
Bx
Example 11 (Cont.)Using the Component Method to solve for the Resultant.Write each vector in i,j notation.
Ax = 20 m, Ay = 0
A = 20 i
Bx = 40 cos 60o = 20 m
By= 40 sin 60o = +34.6 m
B = 20 i + 34.6 j
Cx = 30 cos 30o = 26 m
C = 26 i  15 j
Cy = 30 sin 60o = 15 m
By
30o
B
C
R
Ry
A
60
f
q
Rx
Ax
R= (26)2 + (19.6)2 = 32.6 m
Cx
Bx
R
19.6
26
+19.6
tan f =
f
26
Example 11 (Cont.)The Component MethodAdd algebraically:
A = 20 i
B = 20 i + 34.6 j
C = 26 i  15 j
R= 26 i + 19.6 j
q = 143o
By
30o
B
C
R
Ry
A
60
f
q
Rx
Ax
Cx
Bx
R
+19.6
f
26
Example 11 (Cont.)Find the Resultant.R = 26 i + 19.6 j
The Resultant Displacement of the bike is best given by its polar coordinates R and q.
R = 32.6 m; q = 1430
A = 5 m, 900
B
Cy
B = 12 m, 00
350
A
C = 20 m, 350
C
q
R
A = 0 i + 5.00 j
B = 12 i + 0 j
C = 16.4 i – 11.5 j
28.4 i
 6.47 j
Example 12.Find A + B + C for Vectors Shown below.Ax = 0; Ay = +5 m
Bx = +12 m; By = 0
Cx = (20 m) cos 350
Cy = (20 m) sin 350
R =
First Consider A + B Graphically:
B
B
R
B
A
A
Vector DifferenceFor vectors, signs are indicators of direction. Thus, when a vector is subtracted, the sign (direction) must be changed before adding.
R = A + B
Now A – B: First change sign (direction) of B, then add the negative vector.
B
B
B
A
R’
B
A
A
Vector DifferenceFor vectors, signs are indicators of direction. Thus, when a vector is subtracted, the sign (direction) must be changed before adding.
Subtraction results in a significant difference both in the magnitude and the direction of the resultant vector. (A – B) = A  B
Comparison of addition and subtraction of B
B
B
A
R
R’
B
B
A
A
Addition and SubtractionR = A + B
R’ = A  B
+A
A
+B
B
A 2.43 N
B 7.74 N
Example 13.Given A = 2.4 km, N and B = 7.8 km, N: find A – B and B – A.A  B
B  A
R
R
(2.43 N – 7.74 S)
(7.74 N – 2.43 S)
5.31 km, S
5.31 km, N
R
Ry
Rx = R cosq
q
Ry = R sin q
Rx
Summary for VectorsR
Ry
q
Rx
Summary Continued:Now A – B: First change sign (direction) of B, then add the negative vector.
B
B
B
A
R’
B
A
A
Vector DifferenceFor vectors, signs are indicators of direction. Thus, when a vector is subtracted, the sign (direction) must be changed before adding.