6.2 Angle and Orthogonality in Inner Product
Spaces
In Section 3.2 we defined the notion of “angle” between vector in Rn. In this section we will extend this idea
to general vector spaces. This will enable us to extend the notion of orthogonality as well, thereby setting the
groundwork for a variety of new applications.
Cauchy–Schwarz Inequality
Recall from Formula 20 of Section 3.2 that the angle between two vectors u and v in is
(1)
We were assured that this formula was valid because it followed from the Cauchy–Schwarz inequality
(Theorem 3.2.4) that
(2)
as required for the inverse cosine to be defined. The following generalization of Theorem 3.2.4 will enable us
to define the angle between two vectors in any real inner product space.
THEOREM 6.2.1 Cauchy–Schwarz Inequality
If u and v are vectors in a real inner product space V, then
(3)
Proof We warn you in advance that the proof presented here depends on a clever trick that is not easy to
motivate.
In the case where the two sides of 3 are equal since and are both zero. Thus, we need only
consider the case where . Making this assumption, let
and let t be any real number. Since the positivity axiom states that the inner product of any vector with itself is
nonnegative, it follows that
, This inequality implies that the quadratic polynomial has either no real roots or a repeated real
root. Therefore, its discriminant must satisfy the inequality . Expressing the coefficients ,
and c in terms of the vectors u and v gives or, equivalently,
Taking square roots of both sides and using the fact that and are nonnegative yields
which completes the proof.
The following two alternative forms of the Cauchy–Schwarz inequality are useful to know:
(4)
(5)
The first of these formulas was obtained in the proof of Theorem 6.2.1, and the second is a variation of the
first.
Angle Between Vectors
Our next goal is to define what is meant by the “angle” between vectors in a real inner product space. As the
first step, we leave it for you to use the Cauchy–Schwarz inequality to show that
(6)
This being the case, there is a unique angle in radian measure for which
(7)
(Figure 6.2.1). This enables us to define the angle θ between u and v to be
(8)
Figure 6.2.1
Spaces
In Section 3.2 we defined the notion of “angle” between vector in Rn. In this section we will extend this idea
to general vector spaces. This will enable us to extend the notion of orthogonality as well, thereby setting the
groundwork for a variety of new applications.
Cauchy–Schwarz Inequality
Recall from Formula 20 of Section 3.2 that the angle between two vectors u and v in is
(1)
We were assured that this formula was valid because it followed from the Cauchy–Schwarz inequality
(Theorem 3.2.4) that
(2)
as required for the inverse cosine to be defined. The following generalization of Theorem 3.2.4 will enable us
to define the angle between two vectors in any real inner product space.
THEOREM 6.2.1 Cauchy–Schwarz Inequality
If u and v are vectors in a real inner product space V, then
(3)
Proof We warn you in advance that the proof presented here depends on a clever trick that is not easy to
motivate.
In the case where the two sides of 3 are equal since and are both zero. Thus, we need only
consider the case where . Making this assumption, let
and let t be any real number. Since the positivity axiom states that the inner product of any vector with itself is
nonnegative, it follows that
, This inequality implies that the quadratic polynomial has either no real roots or a repeated real
root. Therefore, its discriminant must satisfy the inequality . Expressing the coefficients ,
and c in terms of the vectors u and v gives or, equivalently,
Taking square roots of both sides and using the fact that and are nonnegative yields
which completes the proof.
The following two alternative forms of the Cauchy–Schwarz inequality are useful to know:
(4)
(5)
The first of these formulas was obtained in the proof of Theorem 6.2.1, and the second is a variation of the
first.
Angle Between Vectors
Our next goal is to define what is meant by the “angle” between vectors in a real inner product space. As the
first step, we leave it for you to use the Cauchy–Schwarz inequality to show that
(6)
This being the case, there is a unique angle in radian measure for which
(7)
(Figure 6.2.1). This enables us to define the angle θ between u and v to be
(8)
Figure 6.2.1
