In the first
two sections of this chapter we looked at vectors in 2-space and 3-space. You probably
noticed that with the exception of the cross product (which is only defined in
3-space) all of the formulas that we had for vectors in 3-space were natural
extensions of the 2-space formulas. In this section we’re going to extend things
out to a much more general setting. We won’t be able to visualize things in a
geometric setting as we did in the previous two sections but things will extend out nicely. In fact, that was why we started in 2-space and 3-space. We wanted
to start out in a setting where we could visualize some of what was going on
before we generalized things into a setting where visualization was a very
difficult thing to do.
So, let’s get things started off
with the following definition.
Definition 1 Given a positive integer n an ordered n-tuple is
a sequence of n real numbers denoted by (a1 , a2
, ... , an ).
The complete set of all ordered n-tuples is called n-space and
is denoted by ℝn.
Definition 2 Suppose u = (u1, u2,
... ,un) and v = (v1, v2, … vn)
are two vectors in ℝn.
(a)
We
say that u and v are equal if,
u1 = v1 u2
= v2 ... un = vn
(b)
The
sum of u and v is defined to be,
u + v = (u1 + v1 , u2 + v2 , …, un + vn)
(c)
The
negative (or additive inverse) of u is defined to be,
−u = (−u1 , −u2 , …, −un)
(d)
The
difference of two vectors is defined to be,
u − v = u + (−v) = (u1 – v1 , u2 – v2 , …, un – vn)
(e)
If c
is any scalar then the scalar multiple of u is defined to be,
cu = (cu1 , cu2 , …, cun)
(f)
The zero
vector in ℝn is denoted by 0 and is defined to be,
0 = (0,0,…,0)
The basic
properties of arithmetic are still valid in ℝn
so let’s also give those so that we can say that we’ve done that.
Theorem 1 Suppose u = (u1 , u2 , …, un) , v = (v1 , v2 , …, vn) and w = (w1 , w2 , …, wn) are vectors in ℝn and c and
k are scalars then,
(a)
u + v = v + u
(b)
u + (v + w) = (u + v) + w
(c)
u + 0 = 0 + u = u
(d)
u −u = u + (−u) = 0
(e)
1u
= u
(f)
(ck )u = c (ku) = k (cu)
(g)
(c + k )u = cu + ku
(h)
c (u + v) = cu + cv
The proof of
all of these come directly from the definitions above and so won’t be given
here.
We now need to
extend the dot product we saw in the previous section to ℝn and we’ll be giving it a new
name as well.
Definition 3 Suppose u = (u1 , u2 , …, un) and v = (v1 , v2 , …, vn) are two vectors in ℝn then the Euclidean
inner product denoted by u▪v is defined to be:
u▪v = u1 v1 + u2 v2 + ··· + un
vn
So, we can see
that it’s the same notation and is a natural extension to the dot product that
we looked at in
the previous section, we’re just going to call it something different now. In
fact, this is probably the more correct name for it and we should instead say
that we’ve renamed this to the dot product when we were working exclusively in ℝ2 and ℝ3.
Note that when
we add in addition, scalar multiplication and the Euclidean inner product to ℝn we will often call this Euclidean
n-space.
We also have
natural extensions of the properties of the dot product that we saw in the
previous section.
Theorem 2 Suppose u = (u1 , u2 , …, un) , v = (v1 , v2 , …, vn) and w = (w1 , w2 , …, wn) are vectors in ℝn and let c be a scalar
then,
(a)
u▪v = v▪u
(b)
(u + v) ▪w = u▪w + v▪w
(c)
c (u▪v) = (cu) ▪v = u▪ (c v)
(d)
u▪u ≥ 0
(e)
u▪u = 0 if
and only if u = 0.
The proof of
this theorem falls directly from the definition of the Euclidean inner product
and areextensions of proofs given in the previous section and so aren’t given
here.
The final
extension to the work of the previous sections that we need to do is to give
the definition of
the norm for vectors in ℝn and
we’ll use this to define distance in ℝn.
Definition 4 Suppose u = (u1 , u2 , …, un) is a vector in ℝn then the Euclidean norm is,
Definition 5 Suppose u = (u1 , u2 , …, un) and v = (v1 , v2 , …, vn) are two points in ℝn then the Euclidean distance between
them is defined to be,
Notice in this definition that we
called u and v points and then used them as vectors
in the norm. This comes back to the idea that an n-tuple can be thought
of as both a point and a vector and so will often be used interchangeably where
needed.
Just
as we saw in the section on vectors if we have u =1
then we will call u a unit vector and so
the vector u from the previous set of examples is not a unit
vector.
Now
that we’ve gotten both the inner product and the norm taken care of we can give
the following
theorem.
Theorem 3 Suppose u and
v are two vectors in ℝn and θ is the angle between them. Then,
u▪v = ‖u‖ ‖v‖ cos θ
Of
course since we are in ℝn it is hard to visualize just what the angle between the
two vectors is, but provided we can find it we can use this theorem. Also note
that this was the definition of the dot product that we gave in the previous section
and like that section this theorem is most useful for actually determining the angle between two
vectors.
The
proof of this theorem is identical to the proof of Theorem 1 in the previous section
and so isn’t
given here.
The
next theorem is very important and has many uses in the study of vectors. In
fact we’ll need it in the proof of at least one theorem in these notes. The
following theorem is called the Cauchy-Schwarz Inequality.
Theorem 4 Suppose u and
v are two vectors in ℝn then,
│u▪v│≤ ‖u‖ ‖v‖
Here are some
nice properties of the Euclidean norm.
Theorem 5 Suppose u and v are two vectors in ℝn and that c is a scalar
then,
(a)
u ≥ 0
(b)
u = 0 if
and only if u = 0.
(c)
cu = c u
(d)
u + v ≤ u + v - Usually called the Triangle Inequality.
The proof of the first two part
is a direct consequence of the definition of the Euclidean norm and so won’t be
given here.
Here are some
nice properties pertaining to the Euclidean distance.
Theorem 6 Suppose u, v, and w are
vectors in ℝn 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) - Usually called the Triangle
Inequality.
The proof of the first two parts
is a direct consequence of the previous theorem and the proof of the third part
is a direct consequence of the definition of distance and won’t be proven here.
We have one
final topic that needs to be generalized into Euclidean n-space.
Definition
6 Suppose u and v are two vectors in ℝn. We say that u and
v are orthogonal if
u▪v = 0 .
So, this
definition of orthogonality is identical to the definition that we saw when we
were dealing with ℝ2 and ℝ3.
Here is the Pythagorean
Theorem in ℝn.
Theorem 7 Suppose u and v are two orthogonal
vectors in ℝn then,
‖u +v‖2 = ‖u‖2 + ‖v‖2
We’ve got one more theorem that
gives a relationship between the Euclidean inner product and the norm. This may
seem like a silly theorem, but we’ll actually need this theorem towards the end
of the next chapter.
Theorem 8 If u and
v are two vectors in ℝn then,
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