Home » Blog » Compute the dot product of a vector and a vector valued integral

Compute the dot product of a vector and a vector valued integral

by RoRi

Let A = 2 \mathbf{i} - 4 \mathbf{j} + \mathbf{k} and let

    \[ B = \int_0^1 (te^{2t} \mathbf{i} + t \cosh (2t) \mathbf{j} + 2te^{2t} \mathbf{k}) \, dt. \]

Compute the dot product A \cdot B.

First, we evaluate the vector valued integral,

    \begin{align*} B &= \int_0^1 (te^{2t} \mathbf{i} + t \cosh (2t) \mathbf{j} + 2te^{2t} \mathbf{k}) \, dt \\[9pt] &= \left( \frac{1}{4} + \frac{1}{4}e^2, \frac{1}{4} + \frac{1}{2} \sinh 2 - \frac{1}{4} \cosh 2, \frac{1}{2} - \frac{3}{2} e^{-2} \right). \end{align*}

Therefore, the dot product is

    \begin{align*} A \cdot B &= \frac{1}{2} + \frac{1}{2} e^2 - 1 - 2 \sinh 2 + \cosh 2 + \frac{1}{2} - \frac{3}{2} e^{-2} \\[9pt] &= \frac{1}{2}e^2 - e^2 + e^{-2} + \frac{1}{2}e^2 + \frac{1}{2} e^{-2} - \frac{3}{2} e^{-2} \\[9pt] &= 0. \end{align*}

One comment

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):