Post on 01-Apr-2015
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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