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Chapter 11: Problem 3
What is the magnitude of the cross product of two parallel vectors?
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Compute \(\mathbf{r}^{\prime \prime}(t)\) and \(\mathbf{r}^{\prime \prime \prime}(t)\) for the following functions. $$\mathbf{r}(t)=\left\langle 3 t^{12}-t^{2}, t^{8}+t^{3}, t^{-4}-2\right\rangle$$
Properties of dot products Let \(\mathbf{u}=\left\langle u_{1}, u_{2}, u_{3}\right\rangle\) \(\mathbf{v}=\left\langle v_{1}, v_{2}, v_{3}\right\rangle,\) and \(\mathbf{w}=\left\langle w_{1}, w_{2}, w_{3}\right\rangle .\) Prove the following vector properties, where \(c\) is a scalar. $$|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u} \| \mathbf{v}|$$
Evaluate the following definite integrals. $$\int_{1 / 2}^{1}\left(\frac{3}{1+2 t} \mathbf{i}-\pi \csc ^{2}\left(\frac{\pi}{2} t\right) \mathbf{k}\right) d t$$
Compute the indefinite integral of the following functions. $$\mathbf{r}(t)=\langle 2 \cos t, 2 \sin 3 t, 4 \cos 8 t\rangle$$
Cauchy-Schwarz Inequality The definition \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\) implies that \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|\) (because \(|\cos \theta| \leq 1\) ). This inequality, known as the Cauchy-Schwarz Inequality, holds in any number of dimensions and has many consequences. Geometric-arithmetic mean Use the vectors \(\mathbf{u}=\langle\sqrt{a}, \sqrt{b}\rangle\) and \(\mathbf{v}=\langle\sqrt{b}, \sqrt{a}\rangle\) to show that \(\sqrt{a b} \leq(a+b) / 2,\) where \(a \geq 0\) and \(b \geq 0\).
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