- 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
- A vector-valued function $F,$ which is never zero and has a continuous derivative $F'(t)$ for all $t,$ is always parallel to its derivative. Prove that there is a constant vector $A$ and a positive real-valued function $u$ such that $F(t) = u(t)A$ for all $t.$
- By definition, two vectors $A$ and $B$ are parallel if for some nonzero scalar $c,$ $B = cA.$
- If we set $u(t) = e^{ct}$ for some nonzero $c,$ then it follows that $u$ is positive for all $t,$ $F(t)$ is never zero, and $F'(t)$ is parallel to $F(t)$ for all $t.\quad \blacksquare$