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Vectors
Chapter 3Chapter 3
©© 2014 A. Dzyubenko2014 A. Dzyubenko
©© 2014 Brooks/Cole2014 Brooks/Cole
Phys 221Phys 221
[email protected]@csub.edu
http://www.csub.edu/~adzyubenko
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Coordinate SystemsCoordinate Systems
Used to describe the position of a point in spaceCoordinate system consists of
a fixed reference point called the originspecific axes with scales and labelsinstructions on how to label a point relative to the origin and the axes
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Types of Coordinate SystemsTypes of Coordinate Systems
Cartesian Plane polar
x
y
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Cartesian Coordinate SystemCartesian Coordinate System
also called rectangular coordinate systemx- and y-axespoints are labeled (x,y)
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Plane Polar Coordinate SystemPlane Polar Coordinate System
point is distance rfrom the origin in the direction of angle θ, counterclockwise from the positive x axispoints are labeled (r,θ)
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Trigonometry ReviewTrigonometry Review
sideadjacentsideopposite
hypotenusesideadjacent
hypotenusesideopposite
=
=
=
θ
θ
θ
tan
cos
sin
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Conversion between Coordinate Conversion between Coordinate SystemsSystems
⎪⎩
⎪⎨
⎧
+=
=
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tan
yxrxyθ
From Cartesian coordinates to the plane polar coordinates
From the plane polar coordinates to Cartesian coordinates
⎩⎨⎧
==
θθ
sincos
ryrx
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Example 3.1Example 3.1The Cartesian coordinates
are (x,y) = (-3.50, -2.50) m, Find the polar coordinates
Solution:
= +
= − + −
=
2 2
2 2( 3.50 m) ( 2.50 m)4.30 m
r x y
2.50 mtan 0.7143.50 m
216 (signs give quadrant)
yx
θ
θ
−= = =
−= °
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Scalar and Vector Quantities Scalar and Vector Quantities Scalar quantities are completely described by magnitude onlyVector quantities have both magnitude (size) and direction Represented by an arrow, the length of the arrow is proportional to the magnitude of vector
Head of the arrow represents the direction
b
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Vector NotationVector Notation
When handwritten, !use an arrow!:When printed, will be in bold print: AWhen dealing with just the magnitude of a vector in print, an italic letter will be used: A
Ar
A= |A|
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Quick QuizQuick Quiz
Which of the following are vector quantities and which are scalar quantities?(a) your age(b) acceleration(c) velocity(d) speed(e) mass
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Equality of Two VectorsEquality of Two Vectors
BA == ifonlyBAand
if A and B point in the same direction along parallel lines
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Adding Vectors:Adding Vectors:Graphical MethodsGraphical Methods
‘Tip-to-tail’ method: the resultant vector R=A+B is the vector drawn from the tail of A to the tip of B
‘tip-to-tail’ method
‘Tail-to-tail’ (parallelogram) method: the resultant vector R=A+B is the vector drawn from where the tails join, outwards to the opposite corner of the parallelogram
‘tail-to-tail’ method
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Adding Vectors, contAdding Vectors, cont
Add more than two vectors:R=A+B+C+D
R is the vector drawn from the tail of the first vector to the tip of the last vector
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Sum of vectors is independent of the Sum of vectors is independent of the order of the additionorder of the addition
Commutative law of addition:A + B = B + A
Associative law of addition:A + (B+C) = (A+B) + C
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Negative of a VectorNegative of a Vector
Vector -A is negative to vector A if
A + (-A) = 0
that means vectors A and -A have the same magnitude but point in opposite directions
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Subtracting VectorsSubtracting Vectors
Define the operation A-B as vector -B added to vector A
A – B = A + (-B)
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Quick QuizQuick Quiz
If vector B is added to vector A, under what condition does the resultant vector A+Bhave magnitude A+B?(a) A and B are parallel and in the same
direction(b) A and B are parallel and in opposite
directions(c) A and B are perpendicular
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Multiplying a Vector by a ScalarMultiplying a Vector by a Scalar
B = mA ??
B has the same direction as A and B = mA
m > 0
m < 0 B has the opposite direction to A and B = mA
A
B
B
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Components of a VectorComponents of a Vector
The projections of vector along coordinate axes are called the components of the vectorAx, Ay are the components of the vector A:
θθ
sincos
AAAA
y
x
==
The signs of the components Axand Ay depend on the angle θ
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Components of a Vector, contComponents of a Vector, cont
A= Ax+ Ay
⎟⎟⎠
⎞⎜⎜⎝
⎛=
+=
−
x
y
yx
AA
AAA
1
22
tanθ
Any vector can be completely described by its components
The components of a vector can be expressed in any convenientcoordinate system
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Quick QuizQuick QuizChoose the correct response to
make the sentence true:
Component of a vector is (a) always, (b) never, or (c) sometimes larger than the magnitude of the vector
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Unit VectorsUnit Vectors
1kji === |ˆ||ˆ||ˆ|
Dimensionless vectors having a magnitude of exactly 1
Use to specify a given direction Has “hat” on the symbol
Symbols represent unit vectors pointing in the positivex, y and z directions
k,j,i ˆandˆˆ
kji ˆand,ˆ,ˆ
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Unit Vectors, cont.Unit Vectors, cont.
form a set of mutually perpendicular vectors in a right-handed coordinate system
kji ˆand,ˆ,ˆ
xy
z
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UnitUnit--Vector NotationVector Notation
jA
iAˆ
ˆ
yy
xx
A
A
=
=
Vector A lies in the xy plane: A= Ax+ Ay
Consider a point (x,y). It can be specified by the position vector r
jiA ˆˆyx AA +=
jir ˆˆ yx +=
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Vector Addition: Using ComponentsVector Addition: Using ComponentsAdd vector B = (Bx, By) to vector A = (Ax, Ay)The resultant vector R = A+B
jiR
jijiR
ˆ)(ˆ)(
or)ˆˆ()ˆˆ(
yyxx
yxyx
BABA
BBAA
+++=
+++=
The components of the resultant vector jiR ˆˆyx RR +=
yyyxxx BARBAR +=+=
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The magnitude of R: use Pythagorean theorem
( ) ( )2222yyxxyx BABARRR +++=+=
xx
yy
x
y
BABA
RR
++
==θtan
Using ComponentsUsing Components
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ThreeThree--Dimensional VectorsDimensional Vectors
The angle θx that R makes with e.g., the x axis:
( ) ( ) ( )222
222
zzyyxx
zyx
BABABA
RRRR
+++++=
++=
kjiA ˆˆˆzyx AAA ++=
RRxx =θcos
kjiB ˆˆˆzyx BBB ++=
A = (Ax,Ay,Az)B = (Bx,By,Bz)
The sum of A and B is kjiR ˆ)(ˆ)(ˆ)( zzyyxx BABABA +++++=
The magnitude of vector R is
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Ex 3.5 Ex 3.5 –– Taking a HikeTaking a HikeA hiker begins a trip by
first walking 25.0 km southeast from her car. She stops and sets up her tent for the night. On the second day, she walks 40.0 km in a direction 60.0° north of east,
Determine the components of the hiker’s resultant displacement
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Ex 3.5 Ex 3.5 –– SolutionSolutionUse vector components:
cos( 45.0 )(25.0 km)(0.707) = 17.7 km
sin( 45.0 )
(25.0 km)( 0.707) 17.7 km
x
y
A A
A A
= − ° =
= − °
= − = −
cos60.0(40.0 km)(0.500) = 20.0 km
sin60.0
(40.0 km)(0.866) 34.6 km
x
y
B B
B B
= ° =
= °
= =
Find the resultant:
ˆ ˆR = (37.7 + 16.9 ) kmi jr
