Linear Algebra

42 Chapter 1. Linear Systems
2.7 Definition The angle between two nonzero vectors �u, �v ∈ R n is
�u �v
θ = arccos(
��u ���v � )
(the angle between the zero vector and any other vector is defined to be a right
angle).
Thus vectors from R n are orthogonal if and only if their dot product is zero.
2.8 Example These vectors are orthogonal.
� � � �
1 1
−1 1
Although they are shown away from canonical position so that they don’t appear
to touch, nonetheless they are orthogonal.
2.9 Example The R 3 angle formula given at the start of this subsection is a
special case of the definition. Between these two
the angle is
arccos(
� �
1
1
0
(1)(0) + (1)(3) + (0)(2)
� �
0
3
2
√ 1 2 +1 2 +0 2 √ 0 2 +3 2 +2
=0
3
) = arccos( √ √ )
2 2 13
approximately 0.94 radians. Notice that these vectors are not orthogonal. Although
the yz-plane may appear to be perpendicular to the xy-plane, in fact
the two planes are that way only in the weak sense that there are vectors in each
orthogonal to all vectors in the other. Not every vector in each is orthogonal to
all vectors in the other.
Exercises
� 2.10 Find the length of each vector.
� � � � � �
4
3
−1
(a) (b)
(c) 1
1
2
1
(d)
� �
0
0
0
� 2.11 Find the angle between each two, if it is defined.
(e)
⎛ ⎞
1
⎜−1⎟
⎝
1
⎠
0