- 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
- Compute $A \cdot B,$ where $A = 2\,\mathbf{i} - 4\,\mathbf{j} + \mathbf{k}$ and $B = \int_0^2 (te^{2t}\,\mathbf{i} + t\cosh 2t\,\mathbf{j} + 2te^{-t}\,\mathbf{k})\,dt$
- Recall from section 6.18 that $$\text{sinh}\,t = \frac{e^t - e^{-t}}{2};\quad \text{cosh}\,t = \frac{e^t + e^{-t}}{2}$$
- Note that $\text{cosh}\,2t = \frac{e^{2t} + e^{-2t}}{2},$ then:$$B = \int_0^2 (te^{2t}\,\mathbf{i} + \frac{te^{2t} + te^{-2t}}{2}\,\mathbf{j} + 2te^{-2t}\,\mathbf{k})\,dt$$Taking the dot product $A \cdot B,$ we get: $$A \cdot B = \int_0^2 2te^{2t} - 2te^{2t} - 2te^{-2t} + 2te^{-2t}\, dt$$But the terms of the integrand simplify to $0,$ thus $A \cdot B = 0.$