Elementary Vector Analysis

Introduction to Vectors Topic 3 Elementary Vector Analysis

Vectors in ℜ2 and ℜ 3 2

Examples of vectors in ℜ with two entries are w 1  3 0.2 

u =  , v =  , w =    − 1 0.3  w 2  3

Examples of vectors in ℜ with three entries are w 1  0.5  3     u = − 1, v = 1.6 , w = w 2  w 3  0.3   2  Where w1 and w2 are any real numbers.

Sum of Vectors Given two vectors u and v, their sum is the vector u + v obtained by adding corresponding entries of u and v. Example: 2  3 u =   and v =   − 1 3     3 + 2  5  u +v =  =  - 1 + 3  2 

A matrix with only one column is called a column vector, or simply a vector. The entries of a vector are called components. The set of all column vectors with n components is denoted by ℜn

Equivalent Vectors Two vectors are equal if and only if their corresponding entries are equal.

4  7  Thus,   and   are not equal. 7  4 

Scalar Multiple of Vectors Given a vector u and a real number, c, the scalar multiple of u by c is the vector cu obtained by multiplying each entry in u by c. Example: 3 if u =   and c =5  − 1  3   15  Then cu = 5   =    − 1  − 5 

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Vectors: Parentheses Form Sometimes, for convenience (and also to save space) we write a column vector such as  3  in the form (3, -1).  − 1   In this case we use parentheses and comma to distinguish the vector (3, -1) from the 1 x 2 row matrix [3 -1], written with brackets and no comma. Thus

3 − 1 ≠ [3 − 1]  

Vectors: Parentheses Form (3-space) If v = (v1 , v 2 , v3 ) and w = ( w1 , w2 , w3 ) are two vectors in 3-space, the arguments similar to those used for vectors in a plane can be used to establish the following results: Vector v and w are equivalent if and only if v1 = w1 , v2 = w2 , v3 = w3 )

v + w = (v1 + w1 , v 2 + w2 , v3 + w3 )

kv = (kv1 , kv2 , kv3 ) where k is any scalar

Vectors: Parentheses Form (2-space) Equivalent two vectors v = (v1 , v2 ) and w = ( w1 , w2 ) are equivalent if and only if v1 = w1 and v2 = w2

Addition, difference and scalar multiple

two vectors v = (v1 , v2 ) and w = ( w1 , w2 )then addition vector, v + w = (v1 + w1 , v 2 + w 2 ) difference vector, v - w = (v1 - w1 , v 2 - w 2 ) scalar multiple, kv = (kv1 , kv 2 )

Exercise 1 Compute u + v and u- 2v for the following: 1. u = (4, 6), v = (7, 4) 2. u = (- 1, 5, 9), v = (8,- 3,6)

 − 1 − 3  , v =  2 − 1

3. u = 

4. u = 3  , v =  2 

2    8 

 − 1    7 

Geometric Descriptions (2-space)

Geometric Description Vector can be represented geometrically as directed line segments or arrows in 2-space or 3space. B A The direction of the arrow specifies the direction of the vector, and the length of the arrow describes the magnitude of the vector. The tail of the arrow (A) is called initial point of the vector, and the tip of the arrow (B) is the terminal point. So the vector is written as →

v = AB

Let v be any vector in the plane, and assume that v has been positioned so its initial point is at the origin of a rectangular coordinate system. The coordinates of the terminal point of v are called the components of v and it is written as v(v1 , v2 ) y (v1,v2)

x

2

Geometric Descriptions (3-space) Vectors in 3 - p s ace can be described by triples of real numbers by introducing a rectangular coordinate system usually refers to point O and axes x, y and z. y

Equivalent Vectors Vectors with the same length and same direction are called equivalent. Since vector is determined solely by its length and direction, equivalent vectors are regarded as equal even though they may be located in different positions.

S

v R

P equivalent, it is written as v = w. If vector v and w are The vector of a length zero is called the zero vector and denoted by 0.

z

Sum of Vectors

Sum of Vectors

Addition of two vectors will resulting new vector that called unify vector. If v and w are any two vectors, then the sum v + w is the vector determined as follows:

w

v

w+v

v+w

w

Position the vector w so that its initial point coincides with the terminal point of v. The vector v + w is represented by the arrow from the initial point of v to the terminal point of w.

v

v+w

Negative vector (-v) If v is any non zero vector,- vis the negative of vector v. It is defined to have the same magnitude as v but in the opposite direction. This vector has the property v + ( −v ) = 0. v -v

v

v+w=w+v

w

v

w

Q

w

x

Vector Difference (-) If v and w are any two vectors, then the difference of w from v is defined by v −w= v +(−w) there are 2 method to obtain v – w: w

v

method 1: v

w

v-w

method 2: -w

v+(-w)

w

v

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Scalar Multiple (kv)

Exercise 2

If v is a non zero vector and k is a non zero scalar, then the product kv is defined to be the vector whose length is k times the length of v and whose direction is the same as that of v.

Draw the vectors in Exercise 1 with the initial points located at the origin

v 2v

Magnitude of a vector (norm) The length of the vector u is often called the norm or magnitude of u and denoted as

u

The norm of a vector u in 2-space, let u = (u1 , u 2 ) 2

u = u1 + u 2

2

Magnitude of a vector (norm) Example: Given u = (0, 4, 3) and v = (1, -2, 1), find u , v and u + v Solution: u = 0 + 4 + 3 v = 12 + (-2)2 +12 2

The norm of a vector u in 3-space, let u = (u1 , u 2 , u 3 ) 2

2

u = u1 + u 2 + u 3

2

A vector of norm 1 is called a unit vector.

2

2

= 0 + 16 + 9

= 1 + 4 +1

= 25

= 6

=5

u + v = (0 + 1) 2 + (4 + (-2))2 + (3 + 1) 2 = 1 + 4 + 16 = 21

Distance If P1 ( x1 , y1 ) and P2 ( x2 , y 2 ) are two points in 2-space, then the distance d between → the points is the norm of the vector P1 P2 is given by d = ( x 2 − x1 ) 2 + ( y 2 − y1 ) 2 Similarly, if P1 ( x1 , y1 , z 1 ) and P2 ( x2 , y 2 , z 2 ) are points in 3-space, then the distance between them is given by d = ( x 2 − x1 ) 2 + ( y 2 − y1 ) 2 + ( z 2 − z1 ) 2

Exercise 3 For each of the following, find the magnitude of u and v, and the distance between u and v.

1. u = (4, 6), v = (7, 4) 2. u = (- 1, 5, 9), v = (8,- 3,6)

 − 1 − 3    3. u =  2  , v =  − 1  2 3  4. u =   , v =    − 1 2   7  8 

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Dot Product of Vectors Formula to calculate the dot product of two vectors is as below:

u ⋅ v = u1v1 + u 2 v2 + u3v3 for vectors in 3-space u ⋅ v = u1v1 + u 2 v2 for vectors in 2-space Example: Given u= (2,-2), v=(5,8), find u . v Solution: u . v = 2(5) + (-2)(8) = 10 + (-16) = -6

Cross Product of Vectors Cross product will defined a type of vector multiplication that produces a vector as the product, but which applicable only in 3-space. If u = (u1 , u 2 , u 3 )and v = (v1 , v 2 , v3 ) are vectors in 3-space, then the cross product u × v is the vector defined by u × v = (u2v3 − u3v2 , u3v1 − u1v3 , u1v2 − u2 v1 ) or in determinant notation  u2 u × v =   v2

u3 v3

,−

u1 u3 u1 u2   , v1 v3 v1 v2 

Cross Product Example: Find u x v where u = (1, 2, -2) and v = (3, 0, 1).

Angle θ between two vectors If θ is the angle between u and v,

θ v

From the previous example,find the angle θ between vector u and v where u= (2,-2) and v=(5,8) Solution: u . v = 2(5) + (-2)(8) u = 22 + (-2)2 v = 52 +82 = 10 + (-16) = 4 + 4 = 8 = 25+64= 89 = -6

-6 8 89 -6 θ = cos -1 = 102.99o 8 89

cos θ =

Cross Product Instead of memorizing the formula, you can obtain the components of by: u1 u2 u3  v v v   1 2 3 Form the 2 x 3 matrix whose first row contains the components of u and the second row contains the components of v. To find the first component of u x v, delete the first column and take the determinant, to find the second component, delete the second column and take the negative of the determinant and to find the third component, delete the third column and take the determinant.

u×v

Connection between dot product and cross product Dot product and cross product of 2 vectors can be visualized as follows:

1 2 -2 3 0 1

 2 −2 1 −2 1 2  = (2,−7,−6) ,− , u × v =   0 1 3 1 3 0

u

u.v so, cos θ = u v

uxv

u θ

v

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Exercise 4 1.

Find the dot product of the following vector a. b.

u = (2,-4,4), v = (-8,0,4) u = (4,-6,2), v = (8,2,-6)

2. In each question in no 1, find the cosine of the angle θ between u and v 3. Find the cross product of the following vector a. b.

a = (4,6,8), b = (8,-4,2) u =( 3,5,-2), v = (4,2,1)

4. Given u = (2, -2), v = (5,8), and w = (-4,3), find each of the following. a. b. c. d.

u . v (u . v)w u . (2v) ||w||²

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