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
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 R n.
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
equal if
u1 = v1 ,u2 = v2 , …, un = vn
Rn
are called
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 R n .
The zero vector in
0 = (0, 0, …, 0).
Rn
is denoted by 0 and is defined to be the vector
If u = (u1 ,u2 ,…,un) is any vector in R n , 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
Rn
is defined by
v – u = v + (-u) = (v1 – u1 ,v2 – u2 ,…,vn – un)
Theorem 3. 1.1 (Properties of Vector in R n )
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
Theorem 3. 1.2
If v is a vector in
Rn
, and k is a scalar, then
a) 0v = 0
b) k0 = 0
c) (-1) v = - v
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.
3.2 Norm, Dot Product, and Distance in n-space
Definition
Rn
Example
If u = (1,3,-2,7), then in the Euclidean space R4 , the norm of u is
|| u || 12  32  (2) 2  7 2  63  3 7
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
|| v || 22  22  (1) 2  3
Thus,
1
2 2 1
u  (2, 2, 1)  ( , ,  )
3
3 3 3
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
d (u, v) || u  v || (u1  v1 )2  (u2  v2 )2 
 (un  vn )2
Example
If u = (1,3,-2,7) and v = (0,7,2,2), then d(u, v) in R4 is
d (u, v) || u  v || (1  0) 2  (3  7) 2  (2  2) 2  (7  2) 2  58
Dot Product
Rn
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 R n , with the operations of addition, scalar multiplication,
and the Euclidean inner product, as Euclidean n-space.
Theorem 3.2.2 and 3.2.3
If u, v and w are vectors in
a)
b)
c)
d)
e)
f)
g)
h)
i)
Rn
and k is any scalar, then
u·v=v·u
u · (v+ w) = u · v + u · w
k (u · v) = (ku) · v
v · v ≥ 0; Further, v · v = 0 if and only if v = 0
0 · v = v · 0= 0
(u +v) · w = u · w + v · w
u · (v- w) = u · v - u · w
(u -v) · w = u · w - v · w
k (u · v) = u · (kv)
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 R n )
If u = (u1 ,u2 ,…,un) and v = (v1 , v2 ,…,vn) are vectors in
Rn
, then
|u · v| ≤ || u || || v ||
Or in terms of components
| u1v1  u2 v2  ...  un vn | (u12  u22  ...  un2 )1/2 (v12  v22  ...  vn2 )1/2
Properties of Length in
If u and v are vectors in
Rn
Rn
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)
Properties of Distance in R n
If u, v, and w are vectors in
Rn
and k is any scalar, then
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)
Theorem 3.2.7
If u, v, and w are vectors in
Rn
with the Euclidean inner product, then
1
1
2
u  v  || u  v ||  || u  v ||2
4
4
Dot Products as Matrix Multiplication
3.3 Orthogonality
Example
4
In the Euclidean space R , 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.
Example
In 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.
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
Example
Find a point-normal equation of the plane through the point P(-1, 3, -2)
that has normal n=(-2, 1, -1).
Solution:
Lines and Planes Determined by Points and Normals Cont.
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).
Example: Determine whether the given planes are parallel.
4x-y+2z=5 and 7x-3y+4z=8
Solution:
Orthogonal Projections
Theorem 3.3.2 Projection Theorem
If u and a are vectors in Rn, and if ao, 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, and
2. The vector w2 is called the vector component of u orthogonal to a.
Hence w2=u-projau.
In summary,
ua
a
(vector component of u along a)
|| a ||2
ua
u  proja u  u 
a (vector component of u orthogonal to a)
2
|| a ||
projau 
Example
Let 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:
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 ||