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Chapter 11: Problem 35
Find a vector orthogonal to the given vectors. $$\langle 0,1,2\rangle \text { and }\langle-2,0,3\rangle$$
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Let \(\mathbf{u}(t)=\left\langle 1, t, t^{2}\right\rangle, \mathbf{v}(t)=\left\langle t^{2},-2 t, 1\right\rangle\) and \(g(t)=2 \sqrt{t}\). Compute the derivatives of the following functions. $$\mathbf{u}(t) \cdot \mathbf{v}(t)$$
Proof of Product Rule By expressing \(\mathbf{u}\) in terms of its components, prove that $$\frac{d}{d t}(f(t) \mathbf{u}(t))=f^{\prime}(t) \mathbf{u}(t)+f(t) \mathbf{u}^{\prime}(t)$$
Suppose the vector-valued function \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\) is smooth on an interval containing the point \(t_{0} .\) The line tangent to \(\mathbf{r}(t)\) at \(t=t_{0}\) is the line parallel to the tangent vector \(\mathbf{r}^{\prime}\left(t_{0}\right)\) that passes through \(\left(f\left(t_{0}\right), g\left(t_{0}\right), h\left(t_{0}\right)\right) .\) For each of the following functions, find an equation of the line tangent to the curve at \(t=t_{0} .\) Choose an orientation for the line that is the same as the direction of \(\mathbf{r}^{\prime}\). $$\mathbf{r}(t)=\left\langle e^{t}, e^{2 t}, e^{3 t}\right\rangle ; t_{0}=0$$
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. $$\mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u}$$
Motion on a sphere Prove that \(\mathbf{r}\) describes a curve that lies on the surface of a sphere centered at the origin \(\left(x^{2}+y^{2}+z^{2}=a^{2}\right.\) with \(a \geq 0\) ) if and only if \(\mathbf{r}\) and \(\mathbf{r}^{\prime}\) are orthogonal at all points of the curve.
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