Chapter 3

•Vectors in n-space

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

• Orthogonality

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

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

DefinitionIf u = (u1 ,u2 ,…,un), v = (v1 ,v2 ,…, vn) are vectors in , then thedot product u · v is defined by u · v = u1 v1 + u2 v2 +… + un vn

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

a) u · v = v · u

b) (u + v) · w = u · w + v · w

c) (k u) · v = k(u · v)

d) v · v ≥ 0; Further, v · v = 0 if and only if v = 0

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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)

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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)

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 the vectors u = (-2, 3, 1, 4) and v = (1, 2, 0, -1)are orthogonal, since u · v = (-2)(1) + (3)(2) + (1)(0) + (4)(-1) = 0

Theorem 3.3.3 (Pythagorean Theorem in )If u and v are orthogonal vectors in with the Euclidean innerproduct, then

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2 2 2|| || || || || ||u v u v