- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
14.4 Exercises
- If $G = F \cdot F' \times F'',$ prove that $G' = F \cdot F' \times F'''.$
- Recall from section 13.9 that the cross product of two vectors is orthogonal to those two vectors.
- Taking the dot product first, we get: $$F \cdot \left(F' \times F''\right) = F' \cdot \left(F ' \times F''\right) + F \cdot \left(F ' \times F''\right)'$$But since the cross product of two vectors is orthogonal to those two vectors, $F' \cdot \left(F' \times F''\right) = O$ Using this and extending the result of problem (15) to $\left(F' \times F''\right)',$ we see that $$G' = F \cdot \left(F' \times F'''\right) \quad \blacksquare$$