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Math1300:MainPage/EuclideanSpace
Contents
• 1 Euclidean n-space
♦ 1.1 Defining Addition and Scalar Multiplication in
◊ 1.1.1 Theorem (Properties of Addition
and Scalar Multiplication)
• 2 The Dot Product in Euclidean n-space
♦ 2.1 Theorem (Properties of the Dot Product)
♦ 2.2 Theorem (Cauchy-Schwarz)
♦ 2.3 Corollary (Cauchy-Schwarz)
• 3 Length in Euclidean n-space
♦ 3.1 Theorem (Triangle Inequality)
♦ 3.2 Theorem (Pythagorean Theorem)
♦ 3.3 Theorem (Length Properties)
• 4 Distance in Euclidean n-space
♦ 4.1 Theorem (Properties of Distance in Euclidean
n-space)
Euclidean n-space
We have derived many properties of 2-space and 3-space. In those cases we have studied ordered pairs (x0,y0) and
ordered triples (x0,y0,z0). We now wish to generalize this concept to n-tuples. By this we mean that we will have n
coordinates: a typical n-tuple is
the set of all possible n-tuples by
where v is a real number for
We denote
i
Defining Addition and Scalar Multiplication in
We define addition and scalar multiplication in
If
in a manner analogous to 2-space and 3-space:
and r is any real number, then
•
•
and
Theorem (Properties of Addition and Scalar Multiplication)
Let
and
A1:
be vectors in n-space, and let r and s be real numbers (scalars). Then
M1:
is a vector
A2:
M2:
A3: There exists a vector
A4:
is a vector
For every vector
A5:
Contents
M3:
such that
there exists a vector
such that
M4:
M5:
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Math1300:MainPage/EuclideanSpace
Proof: A vector in
may be viewed as a
matrix. The rules for addition and scalar multiplication of
vectors in
and those for matrices are identical. Hence the proofs given for matrices carry through to vectors
unchanged.
The Dot Product in Euclidean n-space
The dot product has been defined for 2-space and 3-space, and it is a straightforward concept to extend to
if
and
then
The length of a vector in n-space is defined analogously to that in 2-space or 3-space:
The proofs for the different parts of the following theorem are virtually unchanged from those for 2-space and
3-space.
Theorem (Properties of the Dot Product)
•
•
•
•
if and only if
•
With the concepts of length and dot product defined, we may prove the following important theorem:
Theorem (Cauchy-Schwarz)
If
and
are vectors in
then
Proof: We want to consider the vector
varies over all real numbers. Then
The vectors
and
are considered fixed while x
We define the function f(x) by
and observe that f(x) is a polynomial of degree 2 and, since
the function has at most one real root
2
(namely, 0). Remember that for any quadratic polynomial ax + bx + c, the roots are
Theorem (Properties of Addition and Scalar Multiplication)
In particular,
2
Math1300:MainPage/EuclideanSpace
if b2 − 4ac > 0, then the quadratic polynomial has two real roots. This is exactly what doesn't happen here, and so
which implies
and
Corollary (Cauchy-Schwarz)
If
and
are vectors in
then
This implies that there is exactly one angle θ with
so that
that is,
Notice that we now have the concept of both length and angle for
even though we don't have the familiar
geometry of 2-space and 3-space. In particular, we have a criterion to test whether or not two vectors are
orthogonal:
if and only if
Length in Euclidean n-space
Theorem (Triangle Inequality)
If
and
are vectors in
then
Proof:
The following figure indicates why this is called the triangle inequality. The length of one side of the triangle is
less than or equal to the sum of the lengths of the other two sides.
Theorem (Cauchy-Schwarz)
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Math1300:MainPage/EuclideanSpace
Theorem (Pythagorean Theorem)
If
and
are vectors in
then
if and only if
The following figure shows why this is called the Pythagorean theorem.
Theorem (Length Properties)
If
is a vector in
•
•
and r is any real number, then
with equality if and only if
Distance in Euclidean n-space
With n > 3 we lose our usual geometric view of Euclidean n-space. All is not lost, however. We can often extend
the ideas present in the cases where
to higher values of n.
The concept of distance in Euclidean n-space is one of those ideas. If
and
are two points, then we can think of the vector from P to Q as a vector using arrow
notation. The same vector in coordinate notation is
and the
length of
satisfies
the distance from P to Q and denote it by d(P,Q). This means
expressed in coordinate notation.
Theorem (Pythagorean Theorem)
We call this same value
where P and Q are
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Math1300:MainPage/EuclideanSpace
Theorem (Properties of Distance in Euclidean n-space)
Let
,
•
•
•
and
be vectors. Then
with equality if and only if
(Triangle inequality)
Theorem (Properties of Distance in Euclidean n-space)
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