Chapter 12: Problem 64
Give a geometric description of the set of points \((x, y, z)\) that lie on the intersection of the sphere \(x^{2}+y^{2}+z^{2}=36\) and the plane \(z=6\)
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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 \(\mathbf{N}\)
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. $$\begin{array}{l} \mathbf{r}(t)=\langle 4+5 t,-2 t, 1+3 t\rangle ;\\\ \mathbf{R}(s)=\langle 10 s, 6+4 s, 4+6 s\rangle \end{array}$$
Let \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\). a. Assume that \(\lim _{t \rightarrow a} \mathbf{r}(t)=\mathbf{L}=\left\langle L_{1}, L_{2}, L_{3}\right\rangle,\) which means that $$\begin{aligned} &\lim _{t \rightarrow a}|\mathbf{r}(t)-\mathbf{L}|=0 . \text { Prove that }\\\ &\lim _{t \rightarrow a} f(t)=L_{1}, \quad \lim _{t \rightarrow a} g(t)=L_{2}, \text { and } \lim _{t \rightarrow a} h(t)=L_{3}. \end{aligned}$$ $$\begin{aligned} &\text { b. Assume that } \lim _{t \rightarrow a} f(t)=L_{1}, \lim _{t \rightarrow a} g(t)=L_{2}, \text { and }\\\ &\lim _{t \rightarrow a} h(t)=L_{3} . \text { Prove that } \lim _{t \rightarrow a} \mathbf{r}(t)=\mathbf{L}=\left\langle L_{1}, L_{2}, L_{3},\right\rangle\\\ &\text { which means that } \lim _{t \rightarrow a}|\mathbf{r}(t)-\mathbf{L}|=0. \end{aligned}$$
Consider an object moving along the circular trajectory \(\mathbf{r}(t)=\langle A \cos \omega t, A \sin \omega t\rangle,\) where \(A\) and \(\omega\) are constants. a. Over what time interval \([0, T]\) does the object traverse the circle once? b. Find the velocity and speed of the object. Is the velocity constant in either direction or magnitude? Is the speed constant? c. Find the acceleration of the object. d. How are the position and velocity related? How are the position and acceleration related? e. Sketch the position, velocity, and acceleration vectors at four different points on the trajectory with \(A=\omega=1\)
Show that the two-dimensional trajectory $$x(t)=u_{0} t+x_{0}\( and \)y(t)=-\frac{g t^{2}}{2}+v_{0} t+y_{0},\( for \)0 \leq t \leq T$$ of an object moving in a gravitational field is a segment of a parabola for some value of \(T>0 .\) Find \(T\) such that \(y(T)=0\)
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