Linear Algebra

Section II. Linear Geometry of n-Space 39
a result familiar from R 2 and R 3 , when generalized to arbitrary R k , supports
the idea that a line is straight and a plane is flat. Specifically, we’ll see how to
do Euclidean geometry in a “plane” by giving a definition of the angle between
two R n vectors in the plane that they generate.
2.1 Definition The length of a vector �v ∈ R n is this.
��v � =
�
v 2 1 + ···+ v2 n
2.2 Remark This is a natural generalization of the Pythagorean Theorem. A
classic discussion is in [Polya].
We can use that definition to derive a formula for the angle between two
vectors. For a model of what to do, consider two vectors in R3 .
�v
Put them in canonical position and, in the plane that they determine, consider
the triangle formed by �u, �v, and�u − �v.
To that triangle, apply the Law of Cosines,
��u − �v � 2 = ��u � 2 + ��v � 2 − 2 ��u ���v � cos θ
where θ is the angle between �u and �v. Expand both sides
(u1 − v1) 2 +(u2 − v2) 2 +(u3 − v3) 2
and simplify.
�u
=(u 2 1 + u 2 2 + u 2 3)+(v 2 1 + v 2 2 + v 2 3) − 2 ��u ���v � cos θ
θ = arccos( u1v1 + u2v2 + u3v3
)
��u ���v �
In higher dimensions no picture suffices but we can make the same argument
analytically. First, the form of the numerator is clear — it comes from the middle
terms of the squares (u1 − v1) 2 ,(u2 − v2) 2 ,etc.
2.3 Definition The dot product (or inner product, orscalar product) of two
n-component real vectors is the linear combination of their components.
�u �v = u1v1 + u2v2 + ···+ unvn