CPE 332 Computer Engineering Mathematics II

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CPE 332Computer Engineering Mathematics II

• Chapter 1 Vector

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Web Site

• http//cpe.rsu.ac.th/ut
• Download Material, Course Notes
• Download Slides
• Download HW/QZSolutions
• Grading
• Announcements
• Resources

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Today Topics

• Period 1
• Course Outlines
• Course Web Site
• Part I Chapter 1 Vector (Review)
• Breaks
• Period II
• Part I Chapter 1 Vector (Review)
• Assignment
• Homework I ?????????????? ??????????????????
• ??????? Sheet ???????? ??? Download ??????????
• Next Week ??? Chapter 2 ?????? Matrix

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CPE 332 T1-56 Wk2
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Definition of Vector
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Definition of Vector
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Notes

• ????????? Vector ???????????????????
  ?????????????? Vector ???????????
• ?????????????????? Scalar
• ?????????? ???????????? Unit Vector
  ??????????????????? Vector ????
• ?????????????? ?????????????? Component ?????
  Coordinate (x,y,z) ??????????????????????????????
  ? Coordinate
• ?????????????? Ratio ???????????????????
• ???????????????????
• ????????????????????? ???????????? Component
  i,j,k ????? x,y,z
• ?????????????? Cosine ??????
• ???????????????????????????? Unit Vector

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Vector Operations

• ????????? Vector ???????????????????????????
• ??????? ???? ??? ?? ??? ??? ?????????????? Scalar
  ?????????????????????????????????????????
• ??? ???-?? ??? Vector ????? Vector
  ???????????????????????????????
• ?????? ???????????????? Multiplication
  ????????????? Product ?????????????????
• Scalar Product (Dot Product ?) ????? Scalar
• Vector Product (Cross Product X) ????? Vector
  ????????????? Vector ???????????

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Addition and Substraction
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???????????????? Plane Geometry
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r
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Component Vector
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Component Vector in Cartesian Coordinate
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Addition-Subtraction using Component Vector and
Position Vector
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• ??????????????? Vector ????????????? Position
  Vector ??????????????????? Position Vector

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Any vectors in Cartesian Coordinates

• Given 2 Points, P(x1,y1,z1) and Q(x2,y2,z2)
• We have OPPQOQ
• Then PQ OQ OP
• PQ x2iy2jz2k x1iy1jz1k
• PQ (x2-x1)i(y2-y1)j(z2-z1)k

Z
Q(x2,y2,z2)
O
Y
P(x1,y1,z1)
X
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Any vectors in Cartesian Coordinates

• Given 2 Points, P(x1,y1,z1) and Q(x2,y2,z2)
• PQ (x2-x1)i(y2-y1)j(z2-z1)k
• Also magnitude or length of vector is the
  distance between those 2 points (Euclidian
  Distance)
• PQ ?(x2-x1)2(y2-y1)2(z2-z1)2

Z
Q(x2,y2,z2)
O
Y
P(x1,y1,z1)
X
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Direction Cosine/Ratio

• Vector ????????????????????????????
• ???? ??????????????? ???? Position Vector
• ?????? ??? Unit Vector ??????????????????????
  Vector ????
• ?????? ????????????? Component Vector
  ?????????????????
• ???????????????????????????????????????????????
• ????????????? ????????????????????????
  ?????????????????? Cosine ?????? ????? Direction
  Cosine

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Direction Cosine

• Position vector OP
• Magnitude equal to OP ?x2y2z2
• Direction cos?icos?jcos?k
• Called Direction Cosine

F3
We have cos?F1/OP cos?F2/OP cos?F3/OP
F2
F1
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Direction Cosine and Direction Ratio
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Direction Cosine and Direction Ratio
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Example

• Given points P1(2,-4,5) and P2(1,3,-2), find the
  vector P1P2 and its magnitude and direction
• OP1 2i-4j5k and OP2 i3j-2k
• P1P2OP2-OP1-i7j-7k
• P1P2 ?14949?99
• Cos ? -1/?99 then ? 95.8 degree
• Cos ? 7/?99 then ? 45.3 degree
• Cos ? -7/?99 then ? 134.7 degree

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Direction Cosine and Direction Ratio
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Scalar Product(DOT)
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Scalar Product (DOT)
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Scalar(Dot) Product
A
?
n
A?nAcos?
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Scalar(Dot) Product

• A?(BC)A?BA?C
• Let A a1ia2ja3k, B b1ib2jb3k
• We have A?B a1b1a2b2a3b3
• Also
• Given Saibj, the equation of line perpendicular
  to this vector is in the form
• axbyc

Line axbyc
Saibj
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DOT Product
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Example

• Find the angle between the vector
• Ai-j-k and B 2ij2k
• We calculate A?B 1.2-1.1-1.2-1
• Also A ?(111)?3
• Also B ?(414)3
• Then Cos ? -1/3?3
• ? 101.1 degrees

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Vector Product (Cross)
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Cross Product
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3 Vector Products
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Examples

• Let A2i3j-k, Bij2k
• A?B 23-2 3
• A?B (61)i-(41)j(2-3)k7i-5j-k
• A?B is orthogonal to both A and B
• Test A?(A?B) (2i3j-k)?(7i-5j-k)
  14-1510
• Test B?(A?B) (ij2k)?(7i-5j-k) 7-5-20

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Plane Equation in 3D

• ?? 2D ???????????????? general Form
• AxByC
• ?? 3D ???????? Plane ???? General Form
• AxByCzD
• D ???????????? ?????????????????????? ?????? D
  ??????? ?????????????????????
• 3x-2y5z 3 ????????? 3x-2y5z 6

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Example 1

• ????????????? Plane 2x3y2z5 ???? unit vector
  ????????????? Plane ???
• ????? 3 ??? ??? A, B, C ??????
• A x0,y0,??????? z5/2 ? A(0,0,2.5)
• B x1,y0, ??????? z(5-2)/2 ? B(1,0,1.5)
• C x0,y1, ??????? z(5-3)/2 ? C(0,1,1)
• Vector AB x AC ????? Vector ????????????? Plane

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• ?????????????? Vector ??????? multiple ???
  2i3j2k ???????????? Plane 2x3y2zk ????
  ?????? k ???????????????

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Example 2

• ???????????? Plane ????????????? Vector 3i-2j-k
  ?????????????? (1,1,2) ?????? Plane ????
• ??????????????? ?????????????? Plane ????
  3x-2y-z k
• ???????? k ???????????? (1,1,2) ???????????????
  3(1)-2(1)-(2)-1k
• ???????????????????????????? 3x-2y-z10

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HW for Chapter 1 ??????????????

• ??????????? Download ??????? 1
• ??????????? 1 ????????????????????
• Sheet ??? Download ??
• ?????????????????????????????????
• ????????????????????????????