Vectors day 2

Unit vector notation (i,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 i30 m, N = 30 j 30 m, S = -30 j20 m, out = 20 k 20 m, in = -20 k

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

-30 m

+40 m R

f

R = Rxi + Ry j

R = -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.

Example 4 (Cont.): Next we find her displacement in R,q notation.

-30 m

+40 m

R

f

q = 126.9o

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

040tan ; = 59.130

f f

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

q = 1800 – 59.10

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

B2 2(46 km) (35 km)R

R = 57.8 km

46 kmtan35 km

f

f = 52.70 S. of W.

46 km

35 km R = ?

f?

A

R = -46 i – 35 j

q = 232.70

q = 1800 + 52.70

Example 7. Find the components of the 240-N force exerted by the boy on the girl if his arm makes an angle of 280 with the ground.

280

F = 240 NF Fy

Fx

Fy

Fx = -|(240 N) cos 280| = -212 N

Fy = +|(240 N) sin 280| = +113 N

Or in i,j notation:

F = -(212 N)i + (113 N)j

Example 8. Find the components of a 300-N force acting along the handle of a lawn-mower. The angle with the ground is 320.

320

F = 300 N

FFy

Fx

Fy

Fx = -|(300 N) cos 320| = -254 NFy = -|(300 N) sin 320| = -159 N

32o

32o

Or in i,j notation:

F = -(254 N)i - (159 N)j

Component Method1. 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.

3. Write each vector in i,j notation.

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

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.

E

N1. 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.

2 km, E

A

4 km, NB3 km, W

C2 km, S

D

Example 9 (Cont.) Find resultant displacement.3. Write each

vector in i,j notation:A = +2 i

B = + 4 j

C = -3 i

D = - 2 j 4. Add vectors A,B,C,D algebraically to get resultant in i,j notation.

R = -1 i + 2 j

1 km, west and 2 km north of origin.

E

N

2 km, E

A

4 km, NB3 km, W

C2 km, S

D

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

Example 9 (Cont.) Find resultant displacement.

E

N

2 km, E

A

4 km, NB3 km, W

C2 km, S

DResultant Sum is:

R = -1 i + 2 j

Ry= +2 km

Rx = -1 km

R

f

Now, We Find R, q

2 2( 1) (2) 5R

R = 2.24 km

2 kmtan1 km

f

f = 63.40 N or W

Reminder of Significant Units:

E

N

2 kmA

4 kmB3 km

C2 kmDFor convenience, we follow the

practice of assuming three (3) significant figures for all data in problems.

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

Significant Digits for Angles

40 lb

30 lbR

f

q

Ry

Rx

40 lb

30 lbR

q

Ry

Rx

q = 36.9o; 323.1o

Since a tenth of 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:

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

A = 5 m B = 2.1 m

200B

C = 0.5 mR

q

A = 5 m, 00

B = 2.1 m, 200C = 0.5 m, 900

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 ...)

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

Vector f X-component (i)

Y-component (j)

A=5 m

00 + 5 m 0

B=2.1m

200

+(2.1 m) cos 200

+(2.1 m) sin 200

C=.5 m

900

0 + 0.5 m

Rx = Ax+Bx+Cx Ry = Ay+By+Cy

A = 5 m B = 2.1 m

200B

C = 0.5 mR

q

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

Example 10 (Cont.): Find i,j for three vectors: A = 5 m,00; B = 2.1 m, 200; C = 0.5 m, 900.

X-component (i)

Y-component (j)

Ax = + 5.00 m Ay = 0 Bx = +1.97 m By = +0.718 m Cx = 0 Cy = + 0.50 m

A = 5.00 i + 0 j

B = 1.97 i + 0.718 j C = 0 i + 0.50 j

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

R = 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.

2 2(6.97 m) (1.22 m)R

R = 7.08 m

1.22 mtan6.97 m

f q = 9.930 N. of E.

R = 6.97 i + 1.22 j

5. Determine R,q from x,y:

Rx= 6.97 m

Rq

Ry 1.22 m

Diagram for finding R,q:

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?

60o

30o

R

fq

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

A = 20 m, E

B = 40 mC = 30 m

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

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

60o

30o

R

fq

Note: Rx = Ax + Bx + Cx

Ax

B

Bx

Rx

A

C

Cx

Ry = Ay + By + Cy

0

Ry

By

Cy

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

60

30o

Rf

q

Ax

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 mCy = -30 sin 60o = -15 m

B = -20 i + 34.6 j

C = -26 i - 15 j

A = 20 i

Example 11 (Cont.) The Component Method

60

30o

Rf

q

Ax

B

Bx

Rx

A

C

Cx

Ry

By

Cy

Add algebraically:

A = 20 i

B = -20 i + 34.6 j

C = -26 i - 15 j

R = -26 i + 19.6 j

R

-26

+19.6f

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

tan f = 19.6 -26

q = 143o

Example 11 (Cont.) Find the Resultant.

60

30o

Rf

q

Ax

B

Bx

Rx

A

C

Cx

Ry

By

Cy

R = -26 i + 19.6 j

R

-26

+19.6f

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

R = 32.6 m; q = 1430

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

A = 5 m, 900

B = 12 m, 00

C = 20 m, -350

AB

Rq

Ax = 0; Ay = +5 mBx = +12 m; By = 0Cx = (20 m) cos 350Cy = -(20 m) sin -350

A = 0 i + 5.00 j

B = 12 i + 0 j C = 16.4 i – 11.5 j

R = 28.4 i - 6.47 j

C350

Cx

Cy

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

AB

C350

Rq R

qRx = 28.4 m

Ry = -6.47 m

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

6.47 mtan28.4 m

f q = 12.80 S. of E.

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

First Consider A + B Graphically:

B

A

BR = A + B

R

AB

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

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

B

A

B -B

A

-BR’

A

Comparison of addition and subtraction of B

B

A

B

Addition and Subtraction

R = A + B

R

AB -BR’

A

R’ = A - B

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

Example 13. Given A = 2.4 km, N and B = 7.8 km, N: find A – B and B – A.

A 2.43 N

B 7.74 N

A – B; B - A

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

R R

Summary for Vectors A scalar quantity is completely specified by its magnitude

only. (40 m, 10 gal)

A vector quantity is completely specified by its magnitude and direction. (40 m, 300)

Rx

Ry

R

q

Components of R:

Rx = R cos q

Ry = R sin q

Summary Continued:

Rx

Ry

R

q

Resultant of Vectors:

2 2R x y

tan yx

q

Finding the resultant of two perpendicular vectors is like converting from polar (R, q) to the rectangular (Rx, Ry) coordinates.

Component Method for Vectors Start at origin and draw each vector in succession forming a

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

quadrant of resultant. Write each vector in i,j notation (Rx,Ry). Add vectors algebraically to get resultant in i,j notation.

Then convert to (R,q).

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

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

B

A

B -B

A

-BR’

A

Conclusion of Chapter 3B - Vectors