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EXAMPLE 5 Calculating a Scalar Triple Product
Calculate the scalar triple product of the vectors
Solution
From 7,
Remark The symbol makes no sense because we cannot form the cross product of a scalar and a vector. Thus no
ambiguity arises if we write
rather than . However, for clarity we shall usually keep the parentheses.
It follows from 7 that
since the 3 × 3 determinants that represent these products can be obtained from one another by two row interchanges. (Verify.)
These relationships can be remembered by moving the vectors u, v, and w clockwise around the vertices of the triangle in Figure
3.4.6.
Figure 3.4.6
Geometric Interpretation of Determinants
The next theorem provides a useful geometric interpretation of 2 × 2 and 3 × 3 determinants.
THEOREM 3.4.4
