Linear Algebra, 2020a

Section II. Linear Geometry 43
For any nonzero ⃗v, the vector ⃗v/|⃗v| has length one. We say that the second
normalizes ⃗v to length one.
We can use that to get a formula for the angle between two vectors. Consider
two vectors in R 3 where neither is a multiple of the other
⃗v
⃗u
(the special case of multiples will turn out below not to be an exception). They
determine a two-dimensional plane — for instance, put them in canonical position
and take the plane formed by the origin and the endpoints. In that plane consider
the triangle with sides ⃗u, ⃗v, and ⃗u − ⃗v.
Apply the Law of Cosines: |⃗u − ⃗v | 2 = |⃗u | 2 + |⃗v | 2 − 2 |⃗u ||⃗v | cos θ where θ is the
angle between the vectors. The left side gives
(u 1 − v 1 ) 2 +(u 2 − v 2 ) 2 +(u 3 − v 3 ) 2
=(u 2 1 − 2u 1 v 1 + v 2 1)+(u 2 2 − 2u 2 v 2 + v 2 2)+(u 2 3 − 2u 3 v 3 + v 2 3)
while the right side gives this.
(u 2 1 + u 2 2 + u 2 3)+(v 2 1 + v 2 2 + v 2 3)−2 |⃗u ||⃗v | cos θ
Canceling squares u 2 1 , ..., v2 3
and dividing by 2 gives a formula for the angle.
θ = arccos( u 1v 1 + u 2 v 2 + u 3 v 3
|⃗u ||⃗v |
In higher dimensions we cannot draw pictures as above but we can instead
make the argument analytically. First, the form of the numerator is clear; it
comes from the middle terms of (u i − v i ) 2 .
2.3 Definition The dot product (or inner product or scalar product) oftwo
n-component real vectors is the linear combination of their components.
)
⃗u • ⃗v = u 1 v 1 + u 2 v 2 + ···+ u n v n