chapter 4 vector addition when handwritten, use an arrow: when printed, will be in bold print: a...
TRANSCRIPT
![Page 1: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/1.jpg)
Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print:
A When dealing with just the
magnitude of a vector in print, an italic letter will be used: A
A
![Page 2: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/2.jpg)
Chapter 4 Vector Addition Equality of Two Vectors
Two vectors are equal if they have the same magnitude and the same direction
Movement of vectors in a diagram Any vector can be moved parallel to
itself without being affected
![Page 3: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/3.jpg)
Chapter 4 Vector Addition Negative Vectors
Two vectors are negative if they have the same magnitude but are 180° apart (opposite directions)
A = -B
Resultant Vector The resultant vector is the sum of a
given set of vectors
![Page 4: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/4.jpg)
Chapter 4 Vector Addition When adding vectors, their
directions must be taken into account
Units must be the same Graphical Methods
Use scale drawings Algebraic Methods
More convenient
![Page 5: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/5.jpg)
Chapter 4 Vector Addition
The resultant is the sum of two or more vectors. Vectors canbe added by moving the tail of one vector to the head of anothervector without changing the magnitude or direction of the vector.
Note: The red vector R has the same magnitude and direction.
Vector Addition
![Page 6: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/6.jpg)
Chapter 4 Vector Addition
Multiplying a vector by a scalar number changes its lengthbut not its direction unless the scalar is negative.
V 2V -V
![Page 7: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/7.jpg)
Chapter 4 Vector Addition
If two vectors are added at right angles, the magnitudecan be found by using the Pythagorean Theorem
R2 = A2 + B 2 and the angle by Tan
OppAdj
If two vectors are added at any other angle, the magnitudecan be found by the Law of Cosines
and the angle by the Law of Sines
Sin Aa
SinB
b
SinCc
cos2222 ABBAR
![Page 8: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/8.jpg)
Chapter 4 Vector Addition
6 meters
8 meters
10 meters
The distance traveled is 14 meters and the displacementis 10 meters at 36º south of east.
62+82=102
6386
tan1
36°
![Page 9: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/9.jpg)
Chapter 4 Vector Addition
A hiker walks 3 km due east, then makes a 30° turn north of east walks another 5 km. What is the distance and displacement of thehiker?
The distance traveled is 3 km + 5 km = 8 km
R2 = 32+52- 2*3*5*Cos 150°R2 = 9+25+26=60R = 7.7 km
5θsin
7.7150sin
193 km
5 km
The displacement is 7.7 km @ 19° north of east
R
![Page 10: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/10.jpg)
Chapter 4 Vector Addition
Add the following vectors and determine the resultant.3.0 m/s, 45 and 5.0 m/s, 135
5.83 m/s, 104
![Page 11: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/11.jpg)
Chapter 4 Vector Addition
![Page 12: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/12.jpg)
Chapter 4 Vector AdditionA boat travels at 30 m/s due east across a river that is 120 m wideand the current is 12 m/s south. What is the velocity of the boatrelative to shore? How long does it take the boat to cross the river? How far downstream will the boat land?
30 m/s 30 m/s
12 m/s12 m/s
The speed will be 22 3012 = 32. 3 m/s @ 21° downstream.
The time to cross the river will be t = d/v = 120 m / 30 m/s = 4 sThe boat will be d = vt = 12 m/s * 4 s = 48 m downstream.
![Page 13: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/13.jpg)
Chapter 4 Vector Addition
Examples
![Page 14: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/14.jpg)
Chapter 4 Vector Addition
![Page 15: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/15.jpg)
Chapter 4 Vector Addition
![Page 16: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/16.jpg)
Chapter 4 Vector Addition
Add the following vectors and determine the resultant.6.0 m/s, 225 + 2.0 m/s, 90 4.80 m/s, 207.9
![Page 17: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/17.jpg)
Chapter 4 Vector Addition
Add the following vectors and determine the resultant.6.0 m/s, 225 + 2.0 m/s, 90
45°2 m6 m
R
•R2 = 22 + 62 – 2*2*6*cos 45•R2 = 4 + 36 –24 cos 45•R2 = 40 – 16.96 = 23•R = 4.8 m
2
sin
8.4
45sin
17
17
R = 4.8 m @ 208
![Page 18: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/18.jpg)
Chapter 4 Vector Addition
A component is a part
It is useful to use rectangular components These are the
projections of the vector along the x- and y-axes
![Page 19: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/19.jpg)
Chapter 4 Vector Addition The x-component of a vector is the
projection along the x-axis
The y-component of a vector is the projection along the y-axis
Then,
cosAAx
sinAAy
yx AA A
![Page 20: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/20.jpg)
Chapter 4 Vector Addition The previous equations are valid only if θ is
measured with respect to the x-axis The components can be positive or negative
and will have the same units as the original vector
The components are the legs of the right triangle whose hypotenuse is A
May still have to find θ with respect to the positive x-axis
x
y12y
2x A
AtanandAAA
![Page 21: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/21.jpg)
Chapter 4 Vector Addition Choose a coordinate system and
sketch the vectors Find the x- and y-components of all
the vectors Add all the x-components
This gives Rx: xx vR
![Page 22: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/22.jpg)
Chapter 4 Vector Addition Add all the y-components
This gives Ry:
Use the Pythagorean Theorem to find the magnitude of the Resultant:
Use the inverse tangent function to find the direction of R:
yy vR
2y
2x RRR
x
y1
R
Rtan
![Page 23: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/23.jpg)
Chapter 4 Vector Addition
Vector components is taking a vector and finding the correspondinghorizontal and vertical components.
sin
cos
AA
AA
y
x
A
Ay
Ax
Vector resolution
![Page 24: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/24.jpg)
Chapter 4 Vector Addition
A plane travels 500 km at 60°south of east. Find the east and south components of its displacement.
500 km
60°
de
ds
de= 500 km *cos 60°= 250 km
ds= 500 km *sin 60°= 433 km
![Page 25: Chapter 4 Vector Addition When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print,](https://reader036.vdocuments.site/reader036/viewer/2022062320/56649f435503460f94c63eee/html5/thumbnails/25.jpg)
Chapter 4 Vector Addition
Vector equilibrium
Maze Game