Page 299 - Matrix Analysis & Applied Linear Algebra

5.4 Orthogonal Vectors 295

i i
T
In spite of the fact that u v =0, the vectors u = 3 and v = 0 are
1 1
not orthogonal because u v =0.
∗
Now that “right angles” in higher dimensions make sense, how can more
general angles be defined? Proceed just as before, but use the law of cosines
rather than the Pythagorean theorem. Recall that

v

|| v || || u - v ||
u
θ
|| u ||

2
2
2
2
3
the law of cosines in or says u − v = u + v −2 u v cos θ.
If u and v are orthogonal, then this reduces to the Pythagorean theorem. But,
in general,
2 2 2 T T T
u + v − u − v u u + v v − (u − v) (u − v)
cos θ = =
2 u v 2 u v
T
T
2u v u v
= = .
2 u v u v
This easily extends to higher dimensions because if x, y are vectors from any real
inner-product space, then the general CBS inequality (5.3.4) on p. 287 guarantees
that x y / x y is a number in the interval [−1, 1], and hence there is a
unique value θ in [0,π] such that cos θ = x y / x y .

Angles

In a real inner-product space V, the radian measure of the angle be-
tween nonzero vectors x, y ∈V is defined to be the number θ ∈ [0,π]
such that
x y
cos θ = . (5.4.1)
x y