Physics 2211: Lecture 4 Todays Agenda

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Physics 2211 Lecture 4Todays Agenda

• VECTORS (Chapter 3 in your textbook)
• Webassign WA01 extended to this evening (1100
  pm)
• Webassign WA02 dueWed 700 am

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These vectors are all identical
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Vectors (review)

• In 1 dimension, we can specify direction with a
  or - sign.
• In 2 or 3 dimensions, we need more than a sign to
  specify the direction of something
• To illustrate this, consider the position vector
  r in 2 dimensions.

• Example Where is Duluth?
• Choose origin at Atlanta
• Choose coordinates of distance (miles), and
  direction (N,S,E,W)
• In this case r is a vector that points 20
  miles north.

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Vectors...

• An arrow is commonly used to represent a vector
  quantity moreover, there are two common ways to
  symbolize a vector quantity
• Boldface notation A
• Arrow notation

A
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Vectors and their components

• The components of r are its (x,y,z) coordinates
• r (rx ,ry ,rz ) (x,y,z)
• Consider this in 2-D (since its easier to draw)
• rx x r cos ???
• ry y r sin ???

where r r
(x,y)
y
????arctan( y / x )
r
?
x
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Vectors and components

• The magnitude (length) of r is found using the
  Pythagorean theorem

r
y
x

• The length of a vector clearly does not depend on
  its direction.

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Unit Vectors

• A Unit Vector is a vector having length 1 and no
  units
• It is used to specify a direction
• Unit vector u points in the direction of U
• Often denoted with a hat u û
• Useful examples are the Cartesian unit vectors
  i, j, k
• point in the direction of the x, y and z axes

U
y
j
x
i
k
z
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Vector addition

• Consider the vectors A and B. Find A B.

B
B
A
A
A
C A B
B

• We can arrange the vectors as we want, as long as
  we maintain their length and direction!!

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Vector addition using components

• Consider C A B.
• (a) C (Ax i Ay j) (Bx i By j) (Ax
  Bx)i (Ay By)j
• (b) C (Cx i Cy j)
• Comparing components of (a) and (b)
• Cx Ax Bx
• Cy Ay By

By
C
B
Bx
A
Ay
Ax
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Addition is commutative
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Subtraction
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Summary
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Lecture 2, Act 1Vectors

• Vector A 0,2,1
• Vector B 3,0,2
• Vector C 1,-4,2

What is the resultant vector, D, from adding
ABC?
(a) 3,5,-1 (b) 4,-2,5 (c) 5,-2,4
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Lecture 2, Act 1Solution
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
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VECTOR MULTIPLICATION
1. Scalar (Dot) Product
2. Vector (Cross) Product
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Scalar (Dot) Products
Multiplication of vectors that results in a
scalar
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Dot Product (or Scalar Product)
Some properties( a and b are vectors) a?b
b?a q(a?b) (qb)?a b?(qa) (q is a
scalar) a?(b c) (a?b) (a?c) (c is a
vector) The dot product of perpendicular vectors
is 0 !!
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Examples of dot products
i . i j . j k . k 1 i . j j . k k . i
0
Suppose
Then
a 1 i 2 j 3 k b 4 i - 5 j 6 k
a . b 1x4 2x(-5) 3x6 12 a . a 1x1
2x2 3x3 14 b . b 4x4 (-5)x(-5)
6x6 77
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Properties of dot products

• Magnitude
• a2 a2 a . a
• (ax i ay j) . (ax i ay j)
• ax 2(i . i) ay 2(j . j) 2ax ay (i . j)
• ax 2 ay 2
• Pythagorean Theorem!!

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Properties of dot products

• Components
• a ax i ay j az k (ax , ay , az) (a . i,
  a . j, a . k)
• Derivatives

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Vector Product (Cross Product)
Multiplication of vectors that results in a
vector
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The Cross Product

• We can describe the vectorial nature of torque in
  a compact form by introducing the cross
  product.
• The cross product of two vectors is a third
  vector
• A X B C
• The length of C is given by
• C AB sin ?
• The direction of C is perpendicular to the plane
  defined by A and B, and inthe direction defined
  by the right handrule.

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The Cross Product

• Cartesian components of the cross product
• C A X B
• CX AY BZ - BY AZ
• CY AZ BX - BZ AX
• CZ AX BY - BX AY

B
A
C
Note B X A - A X B
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Recap of todays lecture

• VECTORS (CHAPTER 3)
• Vectors and scalars
• Unit vectors
• Vector components
• Adding vectors graphically
• Adding vectors by components
• Example, Act
• Dot and Cross Products