Page 17 - Numerical Methods for Chemical Engineering

6 1 Linear algebra

For example, for the two vectors
   
1 4
5
2
v =   w =   (1.27)
3 6
v · w = v 1 w 1 + v 2 w 2 + v 3 w 3 = (1)(4) + (2)(5) + (3)(6)
= 4 + 10 + 18 = 32 (1.28)
For 3-D vectors, the dot product is proportional to the product of the lengths and the cosine
of the angle between the two vectors,
v · w =|v||w| cos θ (1.29)

where the length of v is
√
|v|= v · v ≥ 0 (1.30)
Therefore, when two vectors are parallel, the magnitude of their dot product is maximal
and equals the product of their lengths, and when two vectors are perpendicular, their dot
product is zero. These ideas carry completely into N- dimensions. The length of a vector
N
v ∈ is

N

√
2
|v|= v · v = v ≥ 0 (1.31)
k
k=1
If v · w = 0, v and w are said to be orthogonal, the extension of the adjective “perpendic-
3
N
ular” from to .If v · w = 0 and |v|=|w|= 1, i.e., both vectors are normalized to
unit length, v and w are said to be orthonormal.
N
The formula for the length |v| of a vector v ∈ satisfies the more general properties
N
of a norm v of v ∈ . A norm v is a rule that assigns a real scalar, v ∈ , to each
N
N
vector v ∈ such that for every v, w ∈ , and for every c ∈ ,wehave
v ≥ 0 0 = 0
v = 0 if and only if (iff) v = 0
(1.32)
cv =|c| v
v + w ≤ v + w
Each norm also provides an accompanying metric, a measure of how different two vectors
are

d(v, w) = v − w (1.33)
In addition to the length, many other possible definitions of norm exist. The p-norm, v p ,
N
of v ∈ is
1/p

N

p
v p = |v k | (1.34)
k=1