Adding Vectors by Components

Dimension

The number of coordinates needed to specify a point(Ex)

(Ex) 0

bull One dimension = position on a linebull Two dimensions = position on a

planebull Three dimensions = position in 3D-

spacebull Four dimensions = 3D + time

To describe a direction

bull x degrees north of east = Start at east and rotate x degrees toward north

bull x degrees east of north = Start at north and rotate x degrees toward east

Revisit Vectors

1) Vectors in one dimension

Fnet =

F1 = 120 N

F2 = 80 N

Dimension

The number of coordinates needed to specify a point(Ex)

(Ex) 0

bull One dimension = position on a linebull Two dimensions = position on a

planebull Three dimensions = position in 3D-

spacebull Four dimensions = 3D + time

To describe a direction

bull x degrees north of east = Start at east and rotate x degrees toward north

bull x degrees east of north = Start at north and rotate x degrees toward east

Revisit Vectors

1) Vectors in one dimension

Fnet =

F1 = 120 N

F2 = 80 N

bull One dimension = position on a linebull Two dimensions = position on a

planebull Three dimensions = position in 3D-

spacebull Four dimensions = 3D + time

To describe a direction

bull x degrees north of east = Start at east and rotate x degrees toward north

bull x degrees east of north = Start at north and rotate x degrees toward east

Revisit Vectors

1) Vectors in one dimension

Fnet =

F1 = 120 N

F2 = 80 N

To describe a direction

bull x degrees north of east = Start at east and rotate x degrees toward north

bull x degrees east of north = Start at north and rotate x degrees toward east

Revisit Vectors

1) Vectors in one dimension

Fnet =

F1 = 120 N

F2 = 80 N

bull x degrees north of east = Start at east and rotate x degrees toward north

bull x degrees east of north = Start at north and rotate x degrees toward east

Revisit Vectors

1) Vectors in one dimension

Fnet =

F1 = 120 N

F2 = 80 N

Revisit Vectors

1) Vectors in one dimension

Fnet =

F1 = 120 N

F2 = 80 N

2) Vectors in two dimensions

Fnet =

Component Vectorsbull

Visualizing Vectors in 2 Dimensions | Mechanics | Khan Academy

bull Vector resolutionndash Breaking a vector into the x- and y-

components

bull Rx =

bull Ry =

ndash vector R =

Rx = the x component of a vector R

Ry = the y component of a vector R

Component Vectorsbull

Visualizing Vectors in 2 Dimensions | Mechanics | Khan Academy

bull Vector resolutionndash Breaking a vector into the x- and y-

components

bull Rx =

bull Ry =

ndash vector R =

Rx = the x component of a vector R

Ry = the y component of a vector R

Examplebull Find the x- and y-component of R if R = 15

cm and ө = 50˚

Adding Vectors by Components

Add the vectors

15 ms

13 ms

40˚

20˚

15 ms

13 ms

40˚

20˚

y-axis

x-axis

resultant vector

40˚

20˚

y-axis

x-axis

40˚

20˚

y-axis

x-axis

15 ms

13 ms

40˚

20˚

y-axis

x-axis

15sin 40˚

15cos40˚

13sin 20˚

13cos20˚

40˚

20˚

y-axis

x-axis

15cos40˚ 13cos20˚

15sin 40˚

13sin 20˚

y-axis

x-axis

15cos40˚ 13cos20˚

15sin 40˚

13sin 20˚resultant

vector

ExampleThe table below represents a set of force vectors These vectors begin at the origin of a coordinate system and end at the coordinates given in the table

a What is the magnitude of the resultant of the sum of these three vectors

b What is the size of the angle ө that the resultant makes with the horizontal (x-axis)

Vector x-value (N) y-value (N)

A 00 60

B 50 00

C 00 -100

bull Find the resultant vector

A (5 7)

B(-8 -3)

C(9 -10)

85

R (6 -6)

Answer

bull Find the resultant vector

A

B

C

3 N

4N

5N

55deg

50deg40deg

25deg

vectors x component (N)

y component (N)

A 3cos 55 = 17 3sin 55 = 25

B 4cos 220 = -31

4sin 220 = -26

C 5cos 310 = 32 5sin 310 = -38

Resultant R 17 - 31 + 32 = 18

25-26-38 = -39

R (18 -39)

43N

ExampleA GPS receiver indicates that your home is 150 km and 400˚ north of west but the only path through the woods leads directly north If you follow the path 50 km before it opens into a field how far and in what direction would you have to walk to reach your homenorth of west = north from west

R 15 km

40˚

5 km A

x km B ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

Ax = 0Bx

Rx = 15 cos 140˚ = - 115

Ay = 5By Ry = 15 sin 140˚ = 96

x km B

ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

- 115 km

46 km

x = 2 2( 115) 46 124

ө + 90˚ = cos-1 (-115124) =158˚ө = 158˚ - 90˚ = 68˚

vectors

x component (km) y component (km)

A 0 5

B Bx By

R 15 cos 140˚ = - 115 15 sin 140˚ = 96

ExampleA football player runs directly down the field for 35 m before turning an additonal 15 m before getting tackled What is the magnitude and direction of the runnerrsquos total displacement

• Adding Vectors by Components
• Dimension
• Slide 3
• To describe a direction
• Slide 5
• Revisit Vectors
• Slide 7
• Component Vectors
• Example
• Adding Vectors by Components (2)
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Example (2)
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Example (3)
• Slide 23
• Slide 24
• Example (4)

Adding Vectors by Components

Add the vectors

15 ms

13 ms

40˚

20˚

15 ms

13 ms

40˚

20˚

y-axis

x-axis

resultant vector

40˚

20˚

y-axis

x-axis

40˚

20˚

y-axis

x-axis

15 ms

13 ms

40˚

20˚

y-axis

x-axis

15sin 40˚

15cos40˚

13sin 20˚

13cos20˚

40˚

20˚

y-axis

x-axis

15cos40˚ 13cos20˚

15sin 40˚

13sin 20˚

y-axis

x-axis

15cos40˚ 13cos20˚

15sin 40˚

13sin 20˚resultant

vector

ExampleThe table below represents a set of force vectors These vectors begin at the origin of a coordinate system and end at the coordinates given in the table

a What is the magnitude of the resultant of the sum of these three vectors

b What is the size of the angle ө that the resultant makes with the horizontal (x-axis)

Vector x-value (N) y-value (N)

A 00 60

B 50 00

C 00 -100

bull Find the resultant vector

A (5 7)

B(-8 -3)

C(9 -10)

85

R (6 -6)

Answer

bull Find the resultant vector

A

B

C

3 N

4N

5N

55deg

50deg40deg

25deg

vectors x component (N)

y component (N)

A 3cos 55 = 17 3sin 55 = 25

B 4cos 220 = -31

4sin 220 = -26

C 5cos 310 = 32 5sin 310 = -38

Resultant R 17 - 31 + 32 = 18

25-26-38 = -39

R (18 -39)

43N

ExampleA GPS receiver indicates that your home is 150 km and 400˚ north of west but the only path through the woods leads directly north If you follow the path 50 km before it opens into a field how far and in what direction would you have to walk to reach your homenorth of west = north from west

R 15 km

40˚

5 km A

x km B ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

Ax = 0Bx

Rx = 15 cos 140˚ = - 115

Ay = 5By Ry = 15 sin 140˚ = 96

x km B

ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

- 115 km

46 km

x = 2 2( 115) 46 124

ө + 90˚ = cos-1 (-115124) =158˚ө = 158˚ - 90˚ = 68˚

vectors

x component (km) y component (km)

A 0 5

B Bx By

R 15 cos 140˚ = - 115 15 sin 140˚ = 96

ExampleA football player runs directly down the field for 35 m before turning an additonal 15 m before getting tackled What is the magnitude and direction of the runnerrsquos total displacement

• Adding Vectors by Components
• Dimension
• Slide 3
• To describe a direction
• Slide 5
• Revisit Vectors
• Slide 7
• Component Vectors
• Example
• Adding Vectors by Components (2)
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Example (2)
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Example (3)
• Slide 23
• Slide 24
• Example (4)

15 ms

13 ms

40˚

20˚

y-axis

x-axis

resultant vector

40˚

20˚

y-axis

x-axis

40˚

20˚

y-axis

x-axis

15 ms

13 ms

40˚

20˚

y-axis

x-axis

15sin 40˚

15cos40˚

13sin 20˚

13cos20˚

40˚

20˚

y-axis

x-axis

15cos40˚ 13cos20˚

15sin 40˚

13sin 20˚

y-axis

x-axis

15cos40˚ 13cos20˚

15sin 40˚

13sin 20˚resultant

vector

ExampleThe table below represents a set of force vectors These vectors begin at the origin of a coordinate system and end at the coordinates given in the table

a What is the magnitude of the resultant of the sum of these three vectors

b What is the size of the angle ө that the resultant makes with the horizontal (x-axis)

Vector x-value (N) y-value (N)

A 00 60

B 50 00

C 00 -100

bull Find the resultant vector

A (5 7)

B(-8 -3)

C(9 -10)

85

R (6 -6)

Answer

bull Find the resultant vector

A

B

C

3 N

4N

5N

55deg

50deg40deg

25deg

vectors x component (N)

y component (N)

A 3cos 55 = 17 3sin 55 = 25

B 4cos 220 = -31

4sin 220 = -26

C 5cos 310 = 32 5sin 310 = -38

Resultant R 17 - 31 + 32 = 18

25-26-38 = -39

R (18 -39)

43N

ExampleA GPS receiver indicates that your home is 150 km and 400˚ north of west but the only path through the woods leads directly north If you follow the path 50 km before it opens into a field how far and in what direction would you have to walk to reach your homenorth of west = north from west

R 15 km

40˚

5 km A

x km B ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

Ax = 0Bx

Rx = 15 cos 140˚ = - 115

Ay = 5By Ry = 15 sin 140˚ = 96

x km B

ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

- 115 km

46 km

x = 2 2( 115) 46 124

ө + 90˚ = cos-1 (-115124) =158˚ө = 158˚ - 90˚ = 68˚

vectors

x component (km) y component (km)

A 0 5

B Bx By

R 15 cos 140˚ = - 115 15 sin 140˚ = 96

ExampleA football player runs directly down the field for 35 m before turning an additonal 15 m before getting tackled What is the magnitude and direction of the runnerrsquos total displacement

• Adding Vectors by Components
• Dimension
• Slide 3
• To describe a direction
• Slide 5
• Revisit Vectors
• Slide 7
• Component Vectors
• Example
• Adding Vectors by Components (2)
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Example (2)
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Example (3)
• Slide 23
• Slide 24
• Example (4)

40˚

20˚

y-axis

x-axis

40˚

20˚

y-axis

x-axis

15 ms

13 ms

40˚

20˚

y-axis

x-axis

15sin 40˚

15cos40˚

13sin 20˚

13cos20˚

40˚

20˚

y-axis

x-axis

15cos40˚ 13cos20˚

15sin 40˚

13sin 20˚

y-axis

x-axis

15cos40˚ 13cos20˚

15sin 40˚

13sin 20˚resultant

vector

ExampleThe table below represents a set of force vectors These vectors begin at the origin of a coordinate system and end at the coordinates given in the table

a What is the magnitude of the resultant of the sum of these three vectors

b What is the size of the angle ө that the resultant makes with the horizontal (x-axis)

Vector x-value (N) y-value (N)

A 00 60

B 50 00

C 00 -100

bull Find the resultant vector

A (5 7)

B(-8 -3)

C(9 -10)

85

R (6 -6)

Answer

bull Find the resultant vector

A

B

C

3 N

4N

5N

55deg

50deg40deg

25deg

vectors x component (N)

y component (N)

A 3cos 55 = 17 3sin 55 = 25

B 4cos 220 = -31

4sin 220 = -26

C 5cos 310 = 32 5sin 310 = -38

Resultant R 17 - 31 + 32 = 18

25-26-38 = -39

R (18 -39)

43N

ExampleA GPS receiver indicates that your home is 150 km and 400˚ north of west but the only path through the woods leads directly north If you follow the path 50 km before it opens into a field how far and in what direction would you have to walk to reach your homenorth of west = north from west

R 15 km

40˚

5 km A

x km B ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

Ax = 0Bx

Rx = 15 cos 140˚ = - 115

Ay = 5By Ry = 15 sin 140˚ = 96

x km B

ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

- 115 km

46 km

x = 2 2( 115) 46 124

ө + 90˚ = cos-1 (-115124) =158˚ө = 158˚ - 90˚ = 68˚

vectors

x component (km) y component (km)

A 0 5

B Bx By

R 15 cos 140˚ = - 115 15 sin 140˚ = 96

ExampleA football player runs directly down the field for 35 m before turning an additonal 15 m before getting tackled What is the magnitude and direction of the runnerrsquos total displacement

• Adding Vectors by Components
• Dimension
• Slide 3
• To describe a direction
• Slide 5
• Revisit Vectors
• Slide 7
• Component Vectors
• Example
• Adding Vectors by Components (2)
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Example (2)
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Example (3)
• Slide 23
• Slide 24
• Example (4)

40˚

20˚

y-axis

x-axis

15 ms

13 ms

40˚

20˚

y-axis

x-axis

15sin 40˚

15cos40˚

13sin 20˚

13cos20˚

40˚

20˚

y-axis

x-axis

15cos40˚ 13cos20˚

15sin 40˚

13sin 20˚

y-axis

x-axis

15cos40˚ 13cos20˚

15sin 40˚

13sin 20˚resultant

vector

ExampleThe table below represents a set of force vectors These vectors begin at the origin of a coordinate system and end at the coordinates given in the table

a What is the magnitude of the resultant of the sum of these three vectors

b What is the size of the angle ө that the resultant makes with the horizontal (x-axis)

Vector x-value (N) y-value (N)

A 00 60

B 50 00

C 00 -100

bull Find the resultant vector

A (5 7)

B(-8 -3)

C(9 -10)

85

R (6 -6)

Answer

bull Find the resultant vector

A

B

C

3 N

4N

5N

55deg

50deg40deg

25deg

vectors x component (N)

y component (N)

A 3cos 55 = 17 3sin 55 = 25

B 4cos 220 = -31

4sin 220 = -26

C 5cos 310 = 32 5sin 310 = -38

Resultant R 17 - 31 + 32 = 18

25-26-38 = -39

R (18 -39)

43N

ExampleA GPS receiver indicates that your home is 150 km and 400˚ north of west but the only path through the woods leads directly north If you follow the path 50 km before it opens into a field how far and in what direction would you have to walk to reach your homenorth of west = north from west

R 15 km

40˚

5 km A

x km B ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

Ax = 0Bx

Rx = 15 cos 140˚ = - 115

Ay = 5By Ry = 15 sin 140˚ = 96

x km B

ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

- 115 km

46 km

x = 2 2( 115) 46 124

ө + 90˚ = cos-1 (-115124) =158˚ө = 158˚ - 90˚ = 68˚

vectors

x component (km) y component (km)

A 0 5

B Bx By

R 15 cos 140˚ = - 115 15 sin 140˚ = 96

ExampleA football player runs directly down the field for 35 m before turning an additonal 15 m before getting tackled What is the magnitude and direction of the runnerrsquos total displacement

• Adding Vectors by Components
• Dimension
• Slide 3
• To describe a direction
• Slide 5
• Revisit Vectors
• Slide 7
• Component Vectors
• Example
• Adding Vectors by Components (2)
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Example (2)
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Example (3)
• Slide 23
• Slide 24
• Example (4)

15 ms

13 ms

40˚

20˚

y-axis

x-axis

15sin 40˚

15cos40˚

13sin 20˚

13cos20˚

40˚

20˚

y-axis

x-axis

15cos40˚ 13cos20˚

15sin 40˚

13sin 20˚

y-axis

x-axis

15cos40˚ 13cos20˚

15sin 40˚

13sin 20˚resultant

vector

ExampleThe table below represents a set of force vectors These vectors begin at the origin of a coordinate system and end at the coordinates given in the table

a What is the magnitude of the resultant of the sum of these three vectors

b What is the size of the angle ө that the resultant makes with the horizontal (x-axis)

Vector x-value (N) y-value (N)

A 00 60

B 50 00

C 00 -100

bull Find the resultant vector

A (5 7)

B(-8 -3)

C(9 -10)

85

R (6 -6)

Answer

bull Find the resultant vector

A

B

C

3 N

4N

5N

55deg

50deg40deg

25deg

vectors x component (N)

y component (N)

A 3cos 55 = 17 3sin 55 = 25

B 4cos 220 = -31

4sin 220 = -26

C 5cos 310 = 32 5sin 310 = -38

Resultant R 17 - 31 + 32 = 18

25-26-38 = -39

R (18 -39)

43N

ExampleA GPS receiver indicates that your home is 150 km and 400˚ north of west but the only path through the woods leads directly north If you follow the path 50 km before it opens into a field how far and in what direction would you have to walk to reach your homenorth of west = north from west

R 15 km

40˚

5 km A

x km B ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

Ax = 0Bx

Rx = 15 cos 140˚ = - 115

Ay = 5By Ry = 15 sin 140˚ = 96

x km B

ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

- 115 km

46 km

x = 2 2( 115) 46 124

ө + 90˚ = cos-1 (-115124) =158˚ө = 158˚ - 90˚ = 68˚

vectors

x component (km) y component (km)

A 0 5

B Bx By

R 15 cos 140˚ = - 115 15 sin 140˚ = 96

ExampleA football player runs directly down the field for 35 m before turning an additonal 15 m before getting tackled What is the magnitude and direction of the runnerrsquos total displacement

• Adding Vectors by Components
• Dimension
• Slide 3
• To describe a direction
• Slide 5
• Revisit Vectors
• Slide 7
• Component Vectors
• Example
• Adding Vectors by Components (2)
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Example (2)
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Example (3)
• Slide 23
• Slide 24
• Example (4)

40˚

20˚

y-axis

x-axis

15cos40˚ 13cos20˚

15sin 40˚

13sin 20˚

y-axis

x-axis

15cos40˚ 13cos20˚

15sin 40˚

13sin 20˚resultant

vector

ExampleThe table below represents a set of force vectors These vectors begin at the origin of a coordinate system and end at the coordinates given in the table

a What is the magnitude of the resultant of the sum of these three vectors

b What is the size of the angle ө that the resultant makes with the horizontal (x-axis)

Vector x-value (N) y-value (N)

A 00 60

B 50 00

C 00 -100

bull Find the resultant vector

A (5 7)

B(-8 -3)

C(9 -10)

85

R (6 -6)

Answer

bull Find the resultant vector

A

B

C

3 N

4N

5N

55deg

50deg40deg

25deg

vectors x component (N)

y component (N)

A 3cos 55 = 17 3sin 55 = 25

B 4cos 220 = -31

4sin 220 = -26

C 5cos 310 = 32 5sin 310 = -38

Resultant R 17 - 31 + 32 = 18

25-26-38 = -39

R (18 -39)

43N

ExampleA GPS receiver indicates that your home is 150 km and 400˚ north of west but the only path through the woods leads directly north If you follow the path 50 km before it opens into a field how far and in what direction would you have to walk to reach your homenorth of west = north from west

R 15 km

40˚

5 km A

x km B ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

Ax = 0Bx

Rx = 15 cos 140˚ = - 115

Ay = 5By Ry = 15 sin 140˚ = 96

x km B

ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

- 115 km

46 km

x = 2 2( 115) 46 124

ө + 90˚ = cos-1 (-115124) =158˚ө = 158˚ - 90˚ = 68˚

vectors

x component (km) y component (km)

A 0 5

B Bx By

R 15 cos 140˚ = - 115 15 sin 140˚ = 96

ExampleA football player runs directly down the field for 35 m before turning an additonal 15 m before getting tackled What is the magnitude and direction of the runnerrsquos total displacement

• Adding Vectors by Components
• Dimension
• Slide 3
• To describe a direction
• Slide 5
• Revisit Vectors
• Slide 7
• Component Vectors
• Example
• Adding Vectors by Components (2)
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Example (2)
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Example (3)
• Slide 23
• Slide 24
• Example (4)

y-axis

x-axis

15cos40˚ 13cos20˚

15sin 40˚

13sin 20˚resultant

vector

ExampleThe table below represents a set of force vectors These vectors begin at the origin of a coordinate system and end at the coordinates given in the table

a What is the magnitude of the resultant of the sum of these three vectors

b What is the size of the angle ө that the resultant makes with the horizontal (x-axis)

Vector x-value (N) y-value (N)

A 00 60

B 50 00

C 00 -100

bull Find the resultant vector

A (5 7)

B(-8 -3)

C(9 -10)

85

R (6 -6)

Answer

bull Find the resultant vector

A

B

C

3 N

4N

5N

55deg

50deg40deg

25deg

vectors x component (N)

y component (N)

A 3cos 55 = 17 3sin 55 = 25

B 4cos 220 = -31

4sin 220 = -26

C 5cos 310 = 32 5sin 310 = -38

Resultant R 17 - 31 + 32 = 18

25-26-38 = -39

R (18 -39)

43N

ExampleA GPS receiver indicates that your home is 150 km and 400˚ north of west but the only path through the woods leads directly north If you follow the path 50 km before it opens into a field how far and in what direction would you have to walk to reach your homenorth of west = north from west

R 15 km

40˚

5 km A

x km B ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

Ax = 0Bx

Rx = 15 cos 140˚ = - 115

Ay = 5By Ry = 15 sin 140˚ = 96

x km B

ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

- 115 km

46 km

x = 2 2( 115) 46 124

ө + 90˚ = cos-1 (-115124) =158˚ө = 158˚ - 90˚ = 68˚

vectors

x component (km) y component (km)

A 0 5

B Bx By

R 15 cos 140˚ = - 115 15 sin 140˚ = 96

ExampleA football player runs directly down the field for 35 m before turning an additonal 15 m before getting tackled What is the magnitude and direction of the runnerrsquos total displacement

• Adding Vectors by Components
• Dimension
• Slide 3
• To describe a direction
• Slide 5
• Revisit Vectors
• Slide 7
• Component Vectors
• Example
• Adding Vectors by Components (2)
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Example (2)
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Example (3)
• Slide 23
• Slide 24
• Example (4)

ExampleThe table below represents a set of force vectors These vectors begin at the origin of a coordinate system and end at the coordinates given in the table

a What is the magnitude of the resultant of the sum of these three vectors

b What is the size of the angle ө that the resultant makes with the horizontal (x-axis)

Vector x-value (N) y-value (N)

A 00 60

B 50 00

C 00 -100

bull Find the resultant vector

A (5 7)

B(-8 -3)

C(9 -10)

85

R (6 -6)

Answer

bull Find the resultant vector

A

B

C

3 N

4N

5N

55deg

50deg40deg

25deg

vectors x component (N)

y component (N)

A 3cos 55 = 17 3sin 55 = 25

B 4cos 220 = -31

4sin 220 = -26

C 5cos 310 = 32 5sin 310 = -38

Resultant R 17 - 31 + 32 = 18

25-26-38 = -39

R (18 -39)

43N

ExampleA GPS receiver indicates that your home is 150 km and 400˚ north of west but the only path through the woods leads directly north If you follow the path 50 km before it opens into a field how far and in what direction would you have to walk to reach your homenorth of west = north from west

R 15 km

40˚

5 km A

x km B ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

Ax = 0Bx

Rx = 15 cos 140˚ = - 115

Ay = 5By Ry = 15 sin 140˚ = 96

x km B

ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

- 115 km

46 km

x = 2 2( 115) 46 124

ө + 90˚ = cos-1 (-115124) =158˚ө = 158˚ - 90˚ = 68˚

vectors

x component (km) y component (km)

A 0 5

B Bx By

R 15 cos 140˚ = - 115 15 sin 140˚ = 96

ExampleA football player runs directly down the field for 35 m before turning an additonal 15 m before getting tackled What is the magnitude and direction of the runnerrsquos total displacement

• Adding Vectors by Components
• Dimension
• Slide 3
• To describe a direction
• Slide 5
• Revisit Vectors
• Slide 7
• Component Vectors
• Example
• Adding Vectors by Components (2)
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Example (2)
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Example (3)
• Slide 23
• Slide 24
• Example (4)

bull Find the resultant vector

A (5 7)

B(-8 -3)

C(9 -10)

85

R (6 -6)

Answer

bull Find the resultant vector

A

B

C

3 N

4N

5N

55deg

50deg40deg

25deg

vectors x component (N)

y component (N)

A 3cos 55 = 17 3sin 55 = 25

B 4cos 220 = -31

4sin 220 = -26

C 5cos 310 = 32 5sin 310 = -38

Resultant R 17 - 31 + 32 = 18

25-26-38 = -39

R (18 -39)

43N

ExampleA GPS receiver indicates that your home is 150 km and 400˚ north of west but the only path through the woods leads directly north If you follow the path 50 km before it opens into a field how far and in what direction would you have to walk to reach your homenorth of west = north from west

R 15 km

40˚

5 km A

x km B ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

Ax = 0Bx

Rx = 15 cos 140˚ = - 115

Ay = 5By Ry = 15 sin 140˚ = 96

x km B

ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

- 115 km

46 km

x = 2 2( 115) 46 124

ө + 90˚ = cos-1 (-115124) =158˚ө = 158˚ - 90˚ = 68˚

vectors

x component (km) y component (km)

A 0 5

B Bx By

R 15 cos 140˚ = - 115 15 sin 140˚ = 96

ExampleA football player runs directly down the field for 35 m before turning an additonal 15 m before getting tackled What is the magnitude and direction of the runnerrsquos total displacement

• Adding Vectors by Components
• Dimension
• Slide 3
• To describe a direction
• Slide 5
• Revisit Vectors
• Slide 7
• Component Vectors
• Example
• Adding Vectors by Components (2)
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Example (2)
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Example (3)
• Slide 23
• Slide 24
• Example (4)

85

R (6 -6)

Answer

bull Find the resultant vector

A

B

C

3 N

4N

5N

55deg

50deg40deg

25deg

vectors x component (N)

y component (N)

A 3cos 55 = 17 3sin 55 = 25

B 4cos 220 = -31

4sin 220 = -26

C 5cos 310 = 32 5sin 310 = -38

Resultant R 17 - 31 + 32 = 18

25-26-38 = -39

R (18 -39)

43N

ExampleA GPS receiver indicates that your home is 150 km and 400˚ north of west but the only path through the woods leads directly north If you follow the path 50 km before it opens into a field how far and in what direction would you have to walk to reach your homenorth of west = north from west

R 15 km

40˚

5 km A

x km B ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

Ax = 0Bx

Rx = 15 cos 140˚ = - 115

Ay = 5By Ry = 15 sin 140˚ = 96

x km B

ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

- 115 km

46 km

x = 2 2( 115) 46 124

ө + 90˚ = cos-1 (-115124) =158˚ө = 158˚ - 90˚ = 68˚

vectors

x component (km) y component (km)

A 0 5

B Bx By

R 15 cos 140˚ = - 115 15 sin 140˚ = 96

ExampleA football player runs directly down the field for 35 m before turning an additonal 15 m before getting tackled What is the magnitude and direction of the runnerrsquos total displacement

• Adding Vectors by Components
• Dimension
• Slide 3
• To describe a direction
• Slide 5
• Revisit Vectors
• Slide 7
• Component Vectors
• Example
• Adding Vectors by Components (2)
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Example (2)
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Example (3)
• Slide 23
• Slide 24
• Example (4)

bull Find the resultant vector

A

B

C

3 N

4N

5N

55deg

50deg40deg

25deg

vectors x component (N)

y component (N)

A 3cos 55 = 17 3sin 55 = 25

B 4cos 220 = -31

4sin 220 = -26

C 5cos 310 = 32 5sin 310 = -38

Resultant R 17 - 31 + 32 = 18

25-26-38 = -39

R (18 -39)

43N

ExampleA GPS receiver indicates that your home is 150 km and 400˚ north of west but the only path through the woods leads directly north If you follow the path 50 km before it opens into a field how far and in what direction would you have to walk to reach your homenorth of west = north from west

R 15 km

40˚

5 km A

x km B ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

Ax = 0Bx

Rx = 15 cos 140˚ = - 115

Ay = 5By Ry = 15 sin 140˚ = 96

x km B

ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

- 115 km

46 km

x = 2 2( 115) 46 124

ө + 90˚ = cos-1 (-115124) =158˚ө = 158˚ - 90˚ = 68˚

vectors

x component (km) y component (km)

A 0 5

B Bx By

R 15 cos 140˚ = - 115 15 sin 140˚ = 96

ExampleA football player runs directly down the field for 35 m before turning an additonal 15 m before getting tackled What is the magnitude and direction of the runnerrsquos total displacement

• Adding Vectors by Components
• Dimension
• Slide 3
• To describe a direction
• Slide 5
• Revisit Vectors
• Slide 7
• Component Vectors
• Example
• Adding Vectors by Components (2)
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Example (2)
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Example (3)
• Slide 23
• Slide 24
• Example (4)

25deg

vectors x component (N)

y component (N)

A 3cos 55 = 17 3sin 55 = 25

B 4cos 220 = -31

4sin 220 = -26

C 5cos 310 = 32 5sin 310 = -38

Resultant R 17 - 31 + 32 = 18

25-26-38 = -39

R (18 -39)

43N

ExampleA GPS receiver indicates that your home is 150 km and 400˚ north of west but the only path through the woods leads directly north If you follow the path 50 km before it opens into a field how far and in what direction would you have to walk to reach your homenorth of west = north from west

R 15 km

40˚

5 km A

x km B ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

Ax = 0Bx

Rx = 15 cos 140˚ = - 115

Ay = 5By Ry = 15 sin 140˚ = 96

x km B

ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

- 115 km

46 km

x = 2 2( 115) 46 124

ө + 90˚ = cos-1 (-115124) =158˚ө = 158˚ - 90˚ = 68˚

vectors

x component (km) y component (km)

A 0 5

B Bx By

R 15 cos 140˚ = - 115 15 sin 140˚ = 96

ExampleA football player runs directly down the field for 35 m before turning an additonal 15 m before getting tackled What is the magnitude and direction of the runnerrsquos total displacement

• Adding Vectors by Components
• Dimension
• Slide 3
• To describe a direction
• Slide 5
• Revisit Vectors
• Slide 7
• Component Vectors
• Example
• Adding Vectors by Components (2)
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Example (2)
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Example (3)
• Slide 23
• Slide 24
• Example (4)

ExampleA GPS receiver indicates that your home is 150 km and 400˚ north of west but the only path through the woods leads directly north If you follow the path 50 km before it opens into a field how far and in what direction would you have to walk to reach your homenorth of west = north from west

R 15 km

40˚

5 km A

x km B ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

Ax = 0Bx

Rx = 15 cos 140˚ = - 115

Ay = 5By Ry = 15 sin 140˚ = 96

x km B

ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

- 115 km

46 km

x = 2 2( 115) 46 124

ө + 90˚ = cos-1 (-115124) =158˚ө = 158˚ - 90˚ = 68˚

vectors

x component (km) y component (km)

A 0 5

B Bx By

R 15 cos 140˚ = - 115 15 sin 140˚ = 96

ExampleA football player runs directly down the field for 35 m before turning an additonal 15 m before getting tackled What is the magnitude and direction of the runnerrsquos total displacement

• Adding Vectors by Components
• Dimension
• Slide 3
• To describe a direction
• Slide 5
• Revisit Vectors
• Slide 7
• Component Vectors
• Example
• Adding Vectors by Components (2)
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Example (2)
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Example (3)
• Slide 23
• Slide 24
• Example (4)

R 15 km

40˚

5 km A

x km B ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

Ax = 0Bx

Rx = 15 cos 140˚ = - 115

Ay = 5By Ry = 15 sin 140˚ = 96

x km B

ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

- 115 km

46 km

x = 2 2( 115) 46 124

ө + 90˚ = cos-1 (-115124) =158˚ө = 158˚ - 90˚ = 68˚

vectors

x component (km) y component (km)

A 0 5

B Bx By

R 15 cos 140˚ = - 115 15 sin 140˚ = 96

ExampleA football player runs directly down the field for 35 m before turning an additonal 15 m before getting tackled What is the magnitude and direction of the runnerrsquos total displacement

• Adding Vectors by Components
• Dimension
• Slide 3
• To describe a direction
• Slide 5
• Revisit Vectors
• Slide 7
• Component Vectors
• Example
• Adding Vectors by Components (2)
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Example (2)
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Example (3)
• Slide 23
• Slide 24
• Example (4)

x km B

ө ˚

Ax + Bx = Rx

0 + Bx = - 115Bx = - 115

Ay + By = Ry

5 + By = 96By = 46

- 115 km

46 km

x = 2 2( 115) 46 124

ө + 90˚ = cos-1 (-115124) =158˚ө = 158˚ - 90˚ = 68˚

vectors

x component (km) y component (km)

A 0 5

B Bx By

R 15 cos 140˚ = - 115 15 sin 140˚ = 96

ExampleA football player runs directly down the field for 35 m before turning an additonal 15 m before getting tackled What is the magnitude and direction of the runnerrsquos total displacement

• Adding Vectors by Components
• Dimension
• Slide 3
• To describe a direction
• Slide 5
• Revisit Vectors
• Slide 7
• Component Vectors
• Example
• Adding Vectors by Components (2)
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Example (2)
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Example (3)
• Slide 23
• Slide 24
• Example (4)

ExampleA football player runs directly down the field for 35 m before turning an additonal 15 m before getting tackled What is the magnitude and direction of the runnerrsquos total displacement

• Adding Vectors by Components
• Dimension
• Slide 3
• To describe a direction
• Slide 5
• Revisit Vectors
• Slide 7
• Component Vectors
• Example
• Adding Vectors by Components (2)
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Example (2)
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Example (3)
• Slide 23
• Slide 24
• Example (4)