- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
14.7 Exercises
- Referring to Exercise 7, let $\mathbf{u}(t)$ denote the unit vector $\mathbf{u}(t) = \sin \omega t\mathbf{i} - \cos \omega t\mathbf{j}.$ Prove that there are constants $A$ and $B$ such that $\mathbf{v} \times \mathbf{a} = A\mathbf{u}(t) + B\,\mathbf{k},$ and express $A$ and $B$ in terms of $a,$ $b,$ and $\omega.$
- Recall from exercise 8: $$\mathbf{v} \times \mathbf{a} = \left(ab\omega^3 \sin \omega t\right)\,\mathbf{i} - \left(ab\omega^3 \cos \omega t\right)\,\mathbf{j} + a^2\omega^3\left(\sin^2 \omega t + \cos^2 \omega t\right)\,\mathbf{k}$$
- We can express the cross product $\mathbf{v} \times \mathbf{a}$ as the linear combination $A\mathbf{u}(t) + B\mathbf{k},$ where $$A = ab\omega^3, \quad B = a^2\omega^3\quad \blacksquare$$