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 ( a 1 ,a 2 ,…,a n ). The set of all ordered n -tuple is called n- space
3. 1 Vectors in n-space
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”
Two vectors u = (u1,u2,…,un) and v = (v1,v2,…, vn) in are called
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)
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:
Theorem 3. 1.2
If v is a vector in , and k is a scalar, then
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
If u = (1,3,-2,7), then in the Euclidean space R4 , the norm of u is
Normalizing a Vector
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.
Find the unit vector u that has the same direction as v = (2, 2, -1).
Solution: The vector v has length
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
If u = (1,3,-2,7) and v = (0,7,2,2), then d(u, v) in R4 is
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
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.
If u, v and w are vectors in and k is any scalar, then
(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
Properties of Distance in
If u, v, and w are vectors in and k is any scalar, then
If u, v, and w are vectors in with the Euclidean inner product, then
Dot Products as Matrix Multiplication
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