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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. Chapter 4 Vector Addition. Equality of Two Vectors - PowerPoint PPT Presentation
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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
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
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
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
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
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
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
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°
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
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
Chapter 4 Vector Addition
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.
Chapter 4 Vector Addition
Examples
Chapter 4 Vector Addition
Chapter 4 Vector Addition
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
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
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
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
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
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
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
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
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
Chapter 4 Vector Addition
Vector equilibrium
Maze Game