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0593_C02_fm Page 37 Monday, May 6, 2002 1:46 PM

Review of Vector Algebra 37

A ⊥
A

L
A

n
FIGURE 2.8.2
Projection of a vector A parallel and
perpendicular to a line.

Observe that the results in Eqs. (2.8.14) and (2.8.15) are identical and thus consistent
with Eq. (2.8.12). Similarly, the results of Eqs. (2.8.16) and (2.8.17) are the same, thus
verifying Eq. (2.8.13). Finally, observe that the results of Eqs. (2.8.14) and (2.8.16) are
different, thus demonstrating the necessity for parentheses on the left sides of Eqs. (2.8.12)
and (2.8.13).
Recall from Eq. (2.6.10) that the projection A of a vector A along a line L is:

A = ( A n n ) (2.8.18)
⋅
||
where n is a unit vector parallel to L, as in Figure 2.8.2. The vector triple product may be
used to obtain A , the component of perpendicular to L. That is,
⊥
A = n ×( A n) (2.8.19)
×
⊥
To see this, use Eqs. (2.8.6) and (2.8.8) to expand the product. That is,

n ×( A n) = A −( n⋅ ) = A A = A ⊥ (2.8.20)
−
×
n
A
||

2.9 Use of the Index Summation Convention

Observe in the previous sections that expressing a vector in terms of mutually perpendic-
ular unit vectors results in a sum of products of the scalar components and the unit vectors.
Specifically, if v is any vector and if n , n , and n are mutually perpendicular unit vectors,
3
1
2
we can express v in the form:
3
v =−v n + v n + v n = v n (2.9.1)
11 2 1 3 3 ∑ i i
= i 1
Because these sums occur so frequently, and because the pattern of the indices is similar
for sums of products, it is convenient to introduce the “summation convention”: if an
index is repeated, there is a sum over the range of the index (usually 1 to 3). This means,
for example, in Eq. (2.9.1), that the summation sign may be deleted. That is, v may be
expressed as:

V = v n (2.9.2)
i i