Chapter 3 Euclidean Vector Spaces

•Vectors in n-space

•Norm, Dot Product, and Distance in n-space

• Orthogonality

•http://www.traileraddict.com/clip/despicable-me/vectors-introduction

DefinitionIf n is a positive integer, then an ordered n-tuple is a sequence of nreal numbers (a1,a2,…,an). The set of all ordered n-tuple is called n-spaceand is denoted by .nR

Note that an ordered n-tuple (a1,a2,…,an) can be viewed either as a “generalized point” or as a “generalized vector”

3. 1 Vectors in n-space

Definition Two vectors u = (u1 ,u2 ,…,un) and v = (v1 ,v2 ,…, vn) in are calledequal if u1 = v1 ,u2 = v2 , …, un = vn

The sum u + v is defined by

u + v = (u1+v1 , u1+v1 , …, un+vn)

and if k is any scalar, the scalar multiple ku is defined by

ku = (ku1 ,ku2 ,…,kun)

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RemarksThe operations of addition and scalar multiplication in this definition are called the standard operations on .nR

If u = (u1 ,u2 ,…,un) is any vector in , then the negative (or additive inverse) of u is denoted by -u and is defined by

-u = (-u1 ,-u2 ,…,-un).

The zero vector in is denoted by 0 and is defined to be the vector 0 = (0, 0, …, 0).

The difference of vectors in is defined by

v – u = v + (-u) = (v1 – u1 ,v2 – u2 ,…,vn – un)

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Theorem 3. 1.1 (Properties of Vector in )If u = (u1 ,u2 ,…,un), v = (v1 ,v2 ,…, vn), and w = (w1 ,w2 ,…,wn) are vectors in and k and m are scalars, then:

a) u + v = v + u

b) u + (v + w) = (u + v) + w

c) u + 0 = 0 + u = u

d) u + (-u) = 0; that is, u – u = 0

e) k(mu) = (km)u

f) k(u + v) = ku + kv

g) (k+m)u = ku+mu

h) 1u = u

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Theorem 3. 1.2 If v is a vector in , and k is a scalar, then

a) 0v = 0

b) k0 = 0

c) (-1) v = - v

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Definition A vector w is a linear combination of the vectors v1, v2,…, vr if it can be expressed in the form

w = k1v1 + k2v2 + · · · + kr vr

where k1, k2, …, kr are scalars. These scalars are called the coefficients of the linear combination.

Note that the linear combination of a single vector is just a scalar multiple of that vector.

Definition

3.2 Norm, Dot Product, and Distance in n-space

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ExampleIf u = (1,3,-2,7), then in the Euclidean space R4 , the norm of u is

2 2 2 2|| || 1 3 ( 2) 7 63 3 7u

DefinitionA vector of norm 1 is called a unit vector. That is, if v is any nonzero vector in Rn , then

Normalizing a Vector

The process of multiplying a nonzero vector by the reciprocal of its length to obtain a unit vector is called normalizing v.

Example:

Find the unit vector u that has the same direction as v = (2, 2, -1).

Solution: The vector v has length

2 2 2|| || 2 2 ( 1) 3v

Thus, 1 2 2 1(2,2, 1) ( , , )3 3 3 3

u

Definition, The standard unit vectors in Rn are:

e1 = (1, 0, … , 0), e2 = (0, 1, …, 0), …, en = (0, 0, …, 1)

In which case every vector v = (v1,v2, …, vn) in Rn can be expressed as

v = (v1,v2, …, vn) = v1e1 + v2e2 +…+ vnen

The distance between the points u = (u1 ,u2 ,…,un) and v = (v1 , v2 ,…,vn) in Rn defined by

2 2 21 1 2 2( , ) || || ( ) ( ) ( )n nd u v u v u v u v u v

ExampleIf u = (1,3,-2,7) and v = (0,7,2,2), then d(u, v) in R4 is

2 2 2 2( , ) || || (1 0) (3 7) ( 2 2) (7 2) 58d u v u v

Distance

Dot Product

ExampleThe dot product of the vectors u = (-1,3,5,7) and v =(5,-4,7,0) in R4 is

u · v = (-1)(5) + (3)(-4) + (5)(7) + (7)(0) = 18

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It is common to refer to , with the operations of addition, scalar multiplication, and the Euclidean inner product, as Euclidean n-space.

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Theorem 3.2.2 and 3.2.3If u, v and w are vectors in and k is any scalar, then

a) u · v = v · ub) u · (v+ w) = u · v + u · wc) k (u · v) = (ku) · vd) v · v ≥ 0; Further, v · v = 0 if and only if v = 0e) 0 · v = v · 0= 0a) (u +v) · w = u · w + v · wb) u · (v- w) = u · v - u · wc) (u -v) · w = u · w - v · wi) k (u · v) = u · (kv)

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Example(3u + 2v) · (4u + v)= (3u) · (4u + v) + (2v) · (4u + v )= (3u) · (4u) + (3u) · v + (2v) · (4u) + (2v) · v=12(u · u) + 11(u · v) + 2(v · v)

Theorem 3.2.4 (Cauchy-Schwarz Inequality in )If u = (u1 ,u2 ,…,un) and v = (v1 , v2 ,…,vn) are vectors in , then |u · v| ≤ || u || || v ||

Or in terms of components

Properties of Length in If u and v are vectors in and k is any scalar, then

a) || u || ≥ 0

b) || u || = 0 if and only if u = 0

c) || ku || = | k ||| u ||

d) || u + v || ≤ || u || + || v || (Triangle inequality for vectors)

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2 2 2 1/2 2 2 2 1/21 1 2 2 1 2 1 2| ... | ( ... ) ( ... )n n n nu v u v u v u u u v v v

a) d(u, v) ≥ 0

b) d(u, v) = 0 if and only if u = v

c) d(u, v) = d(v, u)

d) d(u, v) ≤ d(u, w ) + d(w, v) (Triangle inequality for distances)

Properties of Distance in If u, v, and w are vectors in and k is any scalar, then

Theorem 3.2.7 If u, v, and w are vectors in with the Euclidean inner product, then

2 21 1|| || || ||4 4

u v u v u v

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Dot Products as Matrix Multiplication

3.3 Orthogonality

ExampleIn the Euclidean space , determine if the vectors u = (-2, 3, 1, 4) and v = (1, 2, 0, -1) are orthogonal.Solution: since u · v = (-2)(1) + (3)(2) + (1)(0) + (4)(-1) = 0, u and v are orthogonal.

ExampleIn the Euclidean space R3, determine if the standard unit vectors i=(1, 0, 0), j=(0, 1, 0), k=(0, 0, 1) is an orthogonal set. Solution: we must show that i · j = i ·k = j ·k = 0.

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Lines and Planes Determined by Points and Normals

A line in R2 is determined uniquely by its slope and one of its points, and that a plane in R3 is determined uniquely by its “inclination” and one of its points. One way of specifying slope and inclination is to use a nonzero vector n, called normal, that is orthogonal to the line or plane in question.

The point-normal equation of the line through the point P0(x0, y0) that has normal n=(a, b) is:

a(x-x0)+b(y-y0)=0

The point-normal equation of the plane through the point P0(x0, y0, z0) that has normal n=(a, b, c) is

a(x-x0)+b(y-y0)+c(z-z0)=0

ExampleFind a point-normal equation of the plane through the point P(-1, 3, -2) that has normal n=(-2, 1, -1).

Solution:

Theorem 3.3.1(a)If a and b are constants that are not both zero, then an equation of the form

ax+by+c=0

represents a line in R2 with normal n=(a, b).

(b) If a, b, and c are constant that are not all zero, then an equation of the form

ax+by+cz+d=0 represents a plane in R3 with normal n=(a, b, c).

Lines and Planes Determined by Points and Normals Cont.

Example: Determine whether the given planes are parallel.4x-y+2z=5 and 7x-3y+4z=8

Solution:

Orthogonal Projections

In summary,

(vector component of u along a)2|| ||a

u aproj u a

a

2|| ||a

u au proj u u a

a

(vector component of u orthogonal to a)

Theorem 3.3.2 Projection TheoremIf u and a are vectors in Rn, and if ao, then u can be expressed in exactly one way in the form u=w1+w2, where w1 is a scalar multiple of a and w2 is orthogonal to a.

Note:1.Here the vector w1 is called the orthogonal projection of u on a, or sometimes the vector component of u along a, denoted by projau, and2.The vector w2 is called the vector component of u orthogonal to a. Hence w2=u-projau.

Theorem 3.3.3 (Pythagorean Theorem in Rn)If u and v are orthogonal vectors in Rn with the Euclidean inner product, then 2 2 2|| || || || || ||u v u v

ExampleLet u=(2, -1, 3) and a=(4, -1, 2). Find the vector component of u along a and the vector component of u orthogonal to a. Solution: