1. Vectors in The R2 and R3 Sub Chapter : Terminology
Dot Product and Orthogonal Projection
Cross Product and Its Application
Some Applications :
Computer Graphics
Quantization on Compression Process
Least Square on Optimization
Etc
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2. Vector  besaran yang mempunyai arah Vector notation
Terminology
Vector  besaran yang mempunyai arah
Vector notation
Length (norm) of vector is
Vector satuan  Vector which length (norm) is equal
to one.
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3. Operations of Vectors : Addition two vector (in the same space)
Multiplication of vektor
(a) with scalar
with another vector
Dot Product
Cross Product
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4. be vector in a Vector Space
Addition of vector
Let
and
be vector in a Vector Space
Vector
defined by
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5. Multiplication vektor with scalar
Multiplication of vector
with scalar k,
define as vector which the length is k times of length
of vector . The direction of vector
If k > 0  in the same direction with
If k < 0  opposite direction with
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6. Are vector in the same space
Let
dan
Are vector in the same space
then
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7. Multiplication of two vectors
dot product
cross product
Dot product
Operation of two vectors in the same space
and produce a scalar
Cross product
Operation of two vectors in R3
which produce a orthogonal vector
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8. be in the same space then we have
Dot Product
Let
be in the same space then we have
where
: norm of
 : The angle of two vectors
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9. Find dot product of two vektor dan Answer :
Example 2 :
Find dot product of two vektor
dan
Answer :
Because tan  = 1 or  = 450
= 4
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10. Some properties of dot product : 1. 2. 3.
We get :
Hence
= 2 (2) + 0 (2)
= 4
Some properties of dot product : 1. 2. 3.
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11. DOT PRODUCT Formula can be written as
Formula of dot product can be used to obtain information about the angle () between two vectors
 is acute if and only if u.v > 0
 is obtuse if and only if u.v < 0
 = /2 if and only if u.v = 0
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12. CROSS PRODUCT Definition
Let a =(a1,a2,a3) and b = (b1,b2,b3) are vectors in 3-space, then cross product a x b is the vector defined by
Where i,j,k are standard unit vector
i=(1,0,0), j=(0,1,0) and k=(0,0,1)
Relationships Cross Product and Dot Product
a.(axb) = 0 ( axb ortogonal to a)
b.(axb) = 0 ( axb ortogonal to b)
||axb||2 = ||a||2 ||b||2 – (a.b) (Lagrange Identity)
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13. CROSS PRODUCT Properties of Cross Product
If a,b and c are vectors in 3-space and k : scalar,then
axb = - (bxa)
ax(b+c) = (axb) +(axc)
(a+b)xc = (axc) +(bxc)
k(axb) = (ka)xb = ax(kb)
ax0 = 0xa = 0
axa = 0
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14. CROSS PRODUCT
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15. CROSS PRODUCT
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16. Example :
Find
where
Answer :
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17. CROSS PRODUCT Geometric Interpretation
We can derive formula ||axb|| using Lagrange Identity. The formula is
|| axb || = ||a|| ||b|| sin 
What is this ? a
Area of Parallelogram
||a|| sin
= ||a|| ||b|| sin  
||b|| b
Area of Triangle
= ½ . ||a|| ||b|| sin 
= ½ || axb ||
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18. CROSS PRODUCT
Example
Find the area of triangle determined by the point A(1,2,3), B(2,2,2) and C(2,0,2)
Solution
Let area of ABC triangle C AB
= a = (1,0,-1)
Area of Triangle
= ½ || axb || AC
= b = (1,-2,-1) A B
a x b
= -2i -2k = (-2,0,-2)
|| axb ||
Area of ABC triangle
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19. Example : Let A, B, C be node in R³ : A = (1, –1, –2) B = (4, 1, 0)
Use cross product to find the area of triangular ABC and area of parallelogram ABC!
Answer :
A as initial vector
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20. Area of triangular ABC
Area of parallelogram
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21. Hence, area of triangular :
B as initial vector
= (1,-1,-2) – (4,1,0) = (-3,-2,-2)
= (2,3,3) – (4,1,0) = (-2,2,3)
Hence, area of triangular :
Area of parallelogram
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22. Find cos angle of vectors : a. and b. and
Exercise 4
Find cos angle of vectors :
a and b and
Find orthogonal projection vector a relatively to
vector b.
a dan b Dan 3.
Find :
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23. 4. Find two unit vector which orthogonal to
5. Find a vector which orthogonal to vector
and
6. Find area of triangular PQR, where P (2, 0, –3),
Q (1, 4, 5), and R (7, 2, 9) and find an area of parallelogram !
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24. VECTORS OPERATIONS Let u = (u1,u2) then ||u|| is given by formula
Let u = (u1,u2,u3) then ||u|| is given by formula
Distance between 2 point (vector)
Let A(a1,a2) and B(b1,b2) are two points (vectors) in 2-space, then distance between A and B is given by formula
Let A(a1,a2,a3) and B(b1,b2,b3) are two points (vectors) in 3-space, then distance between A and B is
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25. VECTORS OPERATIONS Example 1 Let u = (1,2,2), ||u||= ? Solution
Determine distance between A(1,1,1) and B(2,3,4)
Solution
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26. VECTORS OPERATIONS
The Equation of plane
Where
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27. Determine the equation of the plane through the point (1,2,1)
Example
Determine the equation of the plane through the point (1,2,1)
and has normal vector (-1,2,3)!
Answer:
To find d, substitute the point (1,2,1)
So we have the equation of the plane is
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28. VECTORS OPERATIONS Distance of the point to the plane
D : distance of point A to the plane
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29. VECTORS OPERATIONS
Distance of the point to the plane
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30. Example
Determine the distance of the point (2,1,1) to field with the equation 3x – y -2z + 5 = 0!
Answer:
Use the formula :
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31. ORTOGONAL PROJECTION
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32. Find ortogonal projection vector
Example
Find ortogonal projection vector
relatively to vector
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33. Answer:
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34. EXERCISES Let a = (k,k,1) and b = (k,3,-4). Find k
a. If angle between a and b is acute
b. If angle between a and b is obtuse
c. If angle between a and b is orthogonal
2. Find orthogonal projection vector a relatively to vector b
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