Linear Algebra, 2020a

46 Chapter One. Linear Systems
2.7 Definition The angle between two nonzero vectors ⃗u,⃗v ∈ R n is
θ = arccos( ⃗u • ⃗v
|⃗u ||⃗v | )
(if either is the zero vector then we take the angle to be a right angle).
2.8 Corollary Vectors from R n are orthogonal, that is, perpendicular, if and only
if their dot product is zero. They are parallel if and only if their dot product
equals the product of their lengths.
2.9 Example These vectors are orthogonal.
( )
1
•
−1
(
1
1
)
= 0
We’ve drawn the arrows away from canonical position but nevertheless the
vectors are orthogonal.
2.10 Example The R 3 angle formula given at the start of this subsection is a
special case of the definition. Between these two
⎛ ⎞
0
⎝3⎠
2
the angle is
⎛ ⎞
1
⎝1⎠
0
(1)(0)+(1)(3)+(0)(2)
arccos( √
12 + 1 2 + 0 2√ 0 2 + 3 2 + 2 )=arccos( 3
√ √ )
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
̌ 2.11 Find the length of each vector.