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 10/11/2018 9:59

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. 10/11/2018 9:59

be vector in a Vector Space Addition of vector Let and be vector in a Vector Space Vector defined by 10/11/2018 9:59

Operations of Vectors : Addition two vector (in the same space) Multiplication of vektor (a) with scalar with another vector Dot Product Cross Product 10/11/2018 9:59

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 10/11/2018 9:59

Are vector in the same space Let dan Are vector in the same space then 10/11/2018 9:59

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 10/11/2018 9:59

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 10/11/2018 9:59

Find dot product of two vektor dan Answer : Example 2 : Find dot product of two vektor dan Answer : Because tan  = 1 or  = 450 = 4 10/11/2018 9:59

Some properties of dot product : 1. 2. 3. We get : Hence = 2 (2) + 0 (2) = 4 Some properties of dot product : 1. 2. 3. 10/11/2018 9:59

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 10/11/2018 9:59

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 10/11/2018 9:59

VECTORS OPERATIONS Example 1 Let u = (1,2,2), ||u||= ? Solution Determine distance between A(1,1,1) and B(2,3,4) Solution 10/11/2018 9:59

ORTOGONAL PROJECTION 10/11/2018 9:59

Find ortogonal projection vector Example Find ortogonal projection vector relatively to vector 10/11/2018 9:59

Answer: 10/11/2018 9:59

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 10/11/2018 9:59

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) 10/11/2018 9:59

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 10/11/2018 9:59

CROSS PRODUCT 10/11/2018 9:59

CROSS PRODUCT 10/11/2018 9:59

Example : Find where Answer : 10/11/2018 9:59

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 || 10/11/2018 9:59

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 10/11/2018 9:59

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 ABCD! Answer : A as initial vector 10/11/2018 9:59

Area of triangular ABC Area of parallelogram 10/11/2018 9:59

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 10/11/2018 9:59 =

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 : 10/11/2018 9:59

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 ! 10/11/2018 9:59