Page 89 - 35Linear Algebra
P. 89
4.3 Directions and Magnitudes 89
Then the Law of Cosines states that:
2
2
2
kv − uk = kuk + kvk − 2kuk kvk cos θ
Then isolate cos θ:
2
2
1
1 2
n 2
n
kv − uk − kuk − kvk 2 = (v − u ) + · · · + (v − u )
1 2
n 2
− (u ) + · · · + (u )
n 2
1 2
− (v ) + · · · + (v )
1 1
n n
= −2u v − · · · − 2u v
Thus,
1 1
n n
kuk kvk cos θ = u v + · · · + u v .
Note that in the above discussion, we have assumed (correctly) that Eu-
n
clidean lengths in R give the usual notion of lengths of vectors for any plane
n
in R . This now motivates the definition of the dot product.
u 1 v 1
.
.
Definition The dot product of u = . and v = . is
.
.
u n v n
n n
1 1
u v := u v + · · · + u v .
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