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Two-Dimensional
Motion and Vectors
Scalars and Vectors
In Physics, quantities are described as eitherscalar quantities or vector
quantities .
A scalar quantityhas only a magnitude (numbers and units) but no
direction.
A vector quantityhas both a magnitude and a direction.
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Which of the follow are vectors?
distance
displacement
mass
weight
temperature
velocity
acceleration
No
Yes
No
Yes
No
Yes
Yes
Vectors...
There are two common ways of indicating that
something is a vector quantity:
Boldface notation:AA
Arrow notation:
AA =
AA
AA
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January 10, 2011 Physics 114A - Lecture 5 5/26
The Components of a VectorThe Components of a VectorLength, angle, and components can be
calculated from each other using trigonometry:
cosxA A q= sinyA A q=
2 2
x yA A A= +
1tan /y xA Aq
-=
January 10, 2011 Physics 114A - Lecture 5 6/26
2D Cartesian and Polar Coordinate Representations2D Cartesian and Polar Coordinate Representations
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Vector addition:
The sum of two vectors is another vector.
A = B + C
B
C A
B
C
Vector subtraction:
Vector subtraction can be defined in
terms of addition.B - C
B
C
B
-C
B - C
= B + (-1)C
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Unit Vectors:
A Unit VectorUnit Vector is a vectorhaving length 1 and no units.
It is used to specify adirection.
Unit vector uu points in thedirection ofUU.
Often denoted with ahat: uu =
UU
x
y
z
ii
jj
kk
l Useful examples are the cartesianunit vectors [ ii, j, k, j, k]
point in the direction of thex, yand zaxes.
R = rxi + ryj + rzk
Vector addition using components:
l Consider CC=AA + BB.
(a) CC = (Axii+ Ayjj) + (Bxii+ Byjj) = (Ax+ Bx)ii+
(Ay+ By)jj
(b) CC = (Cxii+ Cyjj)
l Comparing components of(a) and (b):
Cx= Ax+ Bx
Cy= Ay+ ByCC
BxAA
ByBB
Ax
Ay
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l Vector A = {0,2,1}
l Vector B = {3,0,2}
l Vector C = {1,-4,2}
What is the resultant vector, D, fromadding A+B+C?
(a)(a) {{33,,--44,,22}} (b)(b) {{44,,--22,,55}} (c)(c) {{55,,--22,,44}}
Example
D = (AXi+ AYj+ AZk) + (BXi+ BYj+ BZk) + (CXi+ CYj+ CZk)
= (AX + BX + CX)i+ (AY + BY+ CY)j+ (AZ + BZ + CZ)k
= (0 + 3 + 1)i+ (2 + 0 - 4)j+ (1 + 2 + 2)k
= {4,-2,5}
January 10, 2011 Physics 114A - Lecture 5 12/26
Multiplying VectorsMultiplying Vectors
x y z
x y z
A A i A j A k
B B i B j B k
= + +
= + +
r
r
x x y y z z
AB
A B A B A B A B
A B Cosq
= + +
=
r r
Dot Product (Scalar Product)
Cross Product (Vector Product)
( )
( )
( )
( )
y z z y
z x x z
x y y x
AB
x y z
x y z
A B A B A B i
A B A B j
A B A B k
A B Sin a b
i j k
A A A
B B B
q
= -
+ -
+ -
=
=
r r
(determinant)
Given two vectors:
Note that , ,
and .
A B A A B B
A B B A
^ ^
r r rr r
r rr r
AB is the magnitude of Btimes the projection of Aon B (or vice versa).
Note that AB = BA
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Describing Position in 3-Space A vector is used to establish the position of a particle of
interest. The position vector, r, locates the particle at some
point in time.
January 11, 2011 Physics 114A - Lecture 6 14/2414/24
The Displacement Vector
r xx yy= +r
2 1r r rD = -r r r
2 1
2 2 1 1 ( ) ( )
r r r
x x y y x x y y
xx y y
D = -
= + - +
= D + D
r r r
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Instantaneous Velocity in 3D
V = lim (r / t) as t 0 = dr / dt 3 Components : Vx = dx / dt, etc
Magnitude,
|V| = SQRT( Vx2 + Vy
2 + Vz2)
Average Velocity in 3-D
Vavg = (r2 r1)/(t2-t1)
= r / t
t is scalar so, V vector
parallel to vector
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Properties of VectorsProperties of Vectors
January 10, 2011 Physics 114A - Lecture 5 18/26
The Components of a VectorThe Components of a VectorWe can resolve vector into perpendicular components using
two-dimensional coordinate systems:
Polar Coordinates Cartesian Coordinates
cos25.0 (1.50 m)(0.906) 1.36 mxr r= = =
sin25.0 (1.50 m)(0.423) 0.634 my
r r= = =
2 2 2 2 2(1.36 m) (0.634 m) 2.25 m 1.50 mx yr r r= + = + = =
[ ]1 1tan (0.634 m) / (1.36 m) tan (0.466) 25.0q - -= = =