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Chapter 12: Problem 24
Find an equation or inequality that describes the following objects. A sphere with center (1,2,0) passing through the point (3,4,5)
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A race Two people travel from \(P(4,0)\) to \(Q(-4,0)\) along the paths given by $$ \begin{aligned} \mathbf{r}(t) &=(4 \cos (\pi t / 8), 4 \sin (\pi t / 8)\rangle \text { and } \\\ \mathbf{R}(t) &=\left(4-t,(4-t)^{2}-16\right) \end{aligned} $$ a. Graph both paths between \(P\) and \(Q\) b. Graph the speeds of both people between \(P\) and \(Q\) c. Who arrives at \(Q\) first?
A pair of lines in \(\mathbb{R}^{3}\) are said to be skew if they are neither parallel nor intersecting. Determine whether the following pairs of lines are parallel, intersecting, or skew. If the lines intersect. determine the point(s) of intersection. $$\begin{array}{l} \mathbf{r}(t)=\langle 4+5 t,-2 t, 1+3 t\rangle ;\\\ \mathbf{R}(s)=\langle 10 s, 6+4 s, 4+6 s\rangle \end{array}$$
Find the points (if they exist) at which the following planes and curves intersect. $$z=16 ; \mathbf{r}(t)=\langle t, 2 t, 4+3 t\rangle, \text { for }-\infty < t < \infty$$
Relationship between \(\mathbf{T}, \mathbf{N},\) and a Show that if an object accelerates in the sense that \(d^{2} s / d t^{2}>0\) and \(\kappa \neq 0,\) then the acceleration vector lies between \(\mathbf{T}\) and \(\mathbf{N}\) in the plane of \(\mathbf{T}\) and \(\mathbf{N}\). If an object decelerates in the sense that \(d^{2} s / d t^{2}<0,\) then the acceleration vector lies in the plane of \(\mathbf{T}\) and \(\mathbf{N},\) but not between \(\mathbf{T}\) and \(\mathbf{N}\)
The definition \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\) implies that \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|(\text {because}|\cos \theta| \leq 1) .\) This inequality, known as the Cauchy-Schwarz Inequality, holds in any number of dimensions and has many consequences. 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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