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0593_C02_fm Page 35 Monday, May 6, 2002 1:46 PM

Review of Vector Algebra 35

B

A
FIGURE 2.8.1
A parallelepiped with vectors A, B,
and C along the edges. C
If the vectors A, B, and C coincide with the edges of a parallelepiped as in Figure 2.8.1,
the scalar triple product of A, B, and C is seen to be equal to the volume of the parallel-
epiped. That is, the volume V is:

× ⋅
V = AB C (2.8.6)

Example 2.8.1: Verification of Interchangeability of Terms of Triple Scalar
Products
Let vectors A, B, and C be expressed in terms of mutually perpendicular unit vectors n ,
1
n , and n as:
3
2
A =−3 n − n + n , B = 2 n + 4 n − 7 n , C = − n + 3 n − 5 n (2.8.7)
1 2 3 1 2 3 1 2 3
Verify the equalities of Eq. (2.8.5).
Solution: From Eq. (2.8.2), the vector products A × B and B × C are:

n n n
1 2 3
×
AB = 3 −1 1 = 3 n + 23 n + 14 n (2.8.8)
1 2 3
2 4 −7

n n n
1 2 3
×
BC = 2 4 − = n + 17 n + 10 n (2.8.9)
7
1 2 3
−1 3 −5
Then, from Eq. (2.6.22), the triple scalar products A × B · C and B × C · A are:

×⋅
5
1
23
4
1
3
3
AB C = () − ( ) + ( )( ) + ( ) − ( ) =−4 (2.8.10)
and
×
⋅
1
1
1
0
1
17
BC A = ()( ) + ( ) − ( ) + ()( ) =−4 (2.8.11)
3
The other equalities are verified similarly (see Problem P2.8.1).
Next, the vector triple product has one of the two forms: A × (B × C) or (A × B) × C.
The result is a vector. The position of the parentheses is important, as the two forms
generally produce different results.
To explore this, let the vectors A, B, and C be expressed in terms of mutually perpen-
dicular unit vectors n with scalar components A , B , and C (i = 1, 2, 3). Then, by using
i
i
i
i