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VECTORS
A bee will fly from rose to roseTo gather up some nectar.
And though his path is quite complexHis displacement is a vector!
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WHAT IS A VECTOR?
A vector is a device used when the direction of the parameter being measured is important.Some examples of vectors are: velocity, force, momentum, electric field strength, and torqueA vector quantity is typically written in boldface or by placing an arrow over the label.A vector quantity may be represented by drawing an arrow. The length of the arrow is proportional to the magnitude of the vector quantity and the arrow points in the direction of the vector
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WHAT IS A VECTOR?
The arrow is usually drawn at the origin of a coordinate system and the angle is measured (in radians or degrees) from an axis.Once a vector is drawn, it may be moved to any other location provided the length and direction of the arrow does not change
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VECTOR ADDITION(determining a resultant)
The maximum resultant occurs when the angle between the vectors being added is 00 and the minimum when the angle is 1800
Vectors may be added several different ways: 1. Graphically – draw all vector quantities to
scale, arrange head to tail, connect the origin of the first vector to the head of the last. (REMEMBER THE BEE)
2. Mathematically Law of Cosines - C2= A2 + B2 – 2ABcos q (The
Theory of Pythagoras is a special case) Components (Two and three dimensions coming
soon!)
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DETERMINING COMPONENTSDetermine the components of a velocity vector with magnitude 20 m/s directed at 37o above the + x-axis.
v =20m/s
37o
vy
vx
s
m
s
mv
s
m
s
mv
y
x
1237sin20
16)37cos(20
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EXAMPLE OF VECTOR ADDITION
A
B
C
Cx Ax BxCy Ay By
Ay
Ax Cx
CyBx
By
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VECTOR SUBTRACTION
To determine the difference between two vectors arrange them with the tails togetherThe direction of the resultant is determined by the order of the equation
A
B A - B B - A
A
B
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UNIT VECTORSThis is a fancy name for a vector of
length one (1).A unit vector is constructed by
dividing a vector by its magnitude
kji ˆ,ˆ,ˆ
jiA ˆ3ˆ6
Unit vectors in the x, y, z direction are denoted as:
A unit vector with an x component of 6, a y component of –3, and a z component of zero is written as:
A
AA
ˆ
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UNIT VECTORS - example
If: jiA ˆ3ˆ6
Then the magnitude of A is 6.71 and the unit vector is:
jiA ˆ45.ˆ89.ˆ
22 36 A
^^^
71.6
3
71.6
6jiA
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ADDITION IN UNITVECTOR NOTATION
Adding vectors this way is easy!
kjiA ˆ3ˆ4ˆ8
jiB ˆ5ˆ2
kjiBAC ˆ3ˆˆ6
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OTHER FUNCTIONS WITH VECTORS
Dot product (scalar product) Used to determine the amount of one vector in the direction of another or the angle between vectorsWork is an example of a dot product
(W= F.d (“f dot d”) or, Fdcosq)
A
B
cosABBA
q yyxx BABABA
Using unit vector notation:
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Dot product example
Determine the angle between the following two vectors:
jiB
jiA
ˆ9ˆ2
ˆ4ˆ7
85
65
B
A
yyxx BABABA
0
1
7.47
)85)(65(
50cos
cos)85)(65(50
50)9)(4()2)(7(
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Cross product (vector product) Used to produce a third vector.The result is perpendicular to the first two vectors (See fig. 3 – 20)Torque is an example of a cross product
(t=r x F (“r cross F”) or rFsinq)
OTHER FUNCTIONS WITH VECTORS
A
B
qsinABC
CBA
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Calculating a cross product using a
determinant
This method will require practice!!See TACTIC 5 ON PAGE 51.
YX
YX
ZX
ZX
ZY
ZY
ZYX
ZYX BB
AAk
BB
AAj
BB
AAi
BBB
AAA
kji
BA ˆˆˆ
ˆˆˆ
kABBAjABBAiABBA YXYXXZXZZYZYˆ)(ˆ)(ˆ)(
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DOT AND CROSS PRODUCTS
http://www.tutorvista.com/content/physics/physics-iii/kinematics/scalar-product-animation.php
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VECTORS!KEEP THEM STRAIGHT AND YOUR LIFE WILL
HAVE DIRECTION!