Physics 2211: Lecture 8 Agenda for Today

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Physics 2211 Lecture 8Agenda for Today

• VECTORS More on finding components,
• Vector algebra

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The component representation of a vector
DEPENDS on the coordinate system

• COORDINATE SYSTEM ORIENTATION AND
• LABELS AFFECT
• Component label
• Component sign
• Component magnitude
• HOWEVER
• Components are unaffected by coordinate system
• origin

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Vector components

• It is helpful (but not necessary) to put the tail
  of the
• vector at the origin.

y
A vectors components are PROJECTIONS along
(PARALLEL TO) the coordinate axes
(ax ,ay )
ay
ax
x
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GOOD ADVICE Use graphical representation to
determine sign of components
A vectors components are positive (negative) if
the projections point in the positive (negative)
directions of the coordinate axes
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Using trigonometry to find components

• Finding components from the magnitude and
  direction
• (and vice versa)
• ax a cos ???
• ay a sin ???

????arctan(ay / ax)
y
WARNING It is DANGEROUS to use memorized trig.
formulas to find components
ay
?
ax
x
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The sign of vector components
GOOD ADVICE Use a convenient angle to
find components---insert component sign manually

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Choose the correct vector components for the
coordinate system shown

• Ux U cos ?, Uy U sin ???
• Ux U cos ?, Uy - U sin ???
• Ux -U cos ?, Uy U sin ???
• Ux -U cos ?, Uy -U sin ???
• Ux U sin ?, Uy U cos ???
• Ux U sin ?, Uy -U cos ??
• Ux -U sin ?, Uy U cos ???
• Ux -U sin ?, Uy -U cos ??? ?

y
?
x
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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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Describing a vector using (Cartesian) unit vectors

• Component sign
• Component magnitude
• HOWEVER
• Components are unaffected by coordinate system
• origin

• COORDINATE SYSTEM ORIENTATION AND
• LABELS AFFECT
• Component label
• Component sign
• Component magnitude
• HOWEVER
• Components are unaffected by coordinate system
• origin

• COORDINATE SYSTEM ORIENTATION AND
• LABELS AFFECT
• Component label
• Component sign
• Component magnitude
• HOWEVER
• Components are unaffected by coordinate system
• origin

• COORDINATE SYSTEM ORIENTATION AND
• LABELS AFFECT
• Component label
• Component sign
• Component magnitude
• HOWEVER
• Components are unaffected by coordinate system
• origin

• COORDINATE SYSTEM ORIENTATION AND
• LABELS AFFECT
• Component label
• Component sign
• Component magnitude
• HOWEVER
• Components are unaffected by coordinate system
• origin

10
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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Vectors

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

What is the resultant vector, D, from adding
ABC?
(1) 3,5,-1 (2) 4,-2,5 (3) 5,-2,4
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Solution
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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• 38 degrees South of East
• 13 degrees South of East
• 38 degrees North of East
• 52 degrees North of East

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Recap

• Vectors (Chapter 3)
• Vector Components
• Vector Algebra (addition using
  components)
• For next time Tilted Axes, Changing Coordinate
  Systems (Text 3-4)

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