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Chapter 3

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Chapter 3

- Vectors in n-space
- Norm, Dot Product, and Distance in n-space
- Orthogonality

3. 1 Vectors in n-space

Definition

If n is a positive integer, then an ordered n-tuple is a sequence of n

real numbers (a1,a2,…,an). The set of all ordered n-tuple is called n-space

and is denoted by .

Note that an ordered n-tuple (a1,a2,…,an) can be viewed either as a

“generalized point” or as a “generalized vector”

Definition

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

equal 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)

Remarks

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

The zero vector in is denoted by 0 and is defined to be the vector

0 = (0, 0, …, 0).

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 difference of vectors in is defined by

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

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:

- u + v = v + u
- u + (v + w) = (u + v) + w
- u + 0 = 0 + u = u
- u + (-u) = 0; that is, u – u = 0
- k(mu) = (km)u
- k(u + v) = ku + kv
- (k+m)u = ku+mu
- 1u = u

Theorem 3. 1.2

If v is a vector in , and k is a scalar, then

- 0v = 0
- k0 = 0+ (v + w) = (u + v) + w
- (-1) v = - v

Definition

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

w = k1v1 + k2v2 + · · · + kr vr

where k1, k2, …, krare 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.

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

Definition

Example

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

Normalizing a Vector

Definition

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

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

Thus,

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

Distance

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

Example

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

Dot Product

Definition

If u = (u1,u2,…,un), v = (v1,v2,…, vn) are vectors in , then the

dot product u · v is defined by

u · v = u1v1+ u2v2+… + un vn

Example

The 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

It is common to refer to , with the operations of addition, scalar multiplication,

and the Euclidean inner product, as Euclidean n-space.

Theorem 3.2.2

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

- u · v = v · u
- (u + v) · w = u · w + v · w
- (k u) · v = k(u · v)
- v · v ≥ 0; Further, v · v = 0 if and only if v = 0

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

- || u || ≥ 0
- || u || = 0 if and only if u = 0
- || ku || = | k ||| u ||
- || u + v || ≤ || u || + || v || (Triangle inequality)

Properties of Distance in

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

- d(u, v) ≥ 0
- d(u, v) = 0 if and only if u = v
- d(u, v) = d(v, u)
- d(u, v) ≤ d(u, w ) + d(w, v) (Triangle inequality)

Theorem 3.2.7

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

Dot Products as Matrix Multiplication

3.3 Orthogonality

Example

In 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 inner

product, then