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Chapter 12: Problem 21
Sketch the plane parallel to the \(x y\) -plane through (2,4,2) and find its equation.
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An object moves along a straight line from the point \(P(1,2,4)\) to the point \(Q(-6,8,10)\) a. Find a position function \(\mathbf{r}\) that describes the motion if it occurs with a constant speed over the time interval [0,5] b. Find a position function \(\mathbf{r}\) that describes the motion if it occurs with speed \(e^{t}\)
Consider the trajectory given by the position function $$\mathbf{r}(t)=\left\langle 50 e^{-t} \cos t, 50 e^{-t} \sin t, 5\left(1-e^{-t}\right)\right), \quad \text { for } t \geq 0$$ a. Find the initial point \((t=0)\) and the "terminal" point \(\left(\lim _{t \rightarrow \infty} \mathbf{r}(t)\right)\) of the trajectory. b. At what point on the trajectory is the speed the greatest? c. Graph the trajectory.
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. $$\begin{array}{l} \mathbf{r}(t)=\langle 4+5 t,-2 t, 1+3 t\rangle ;\\\ \mathbf{R}(s)=\langle 10 s, 6+4 s, 4+6 s\rangle \end{array}$$
\(\mathbb{R}^{2}\) Consider the vectors \(\mathbf{I}=\langle 1 / \sqrt{2}, 1 / \sqrt{2}\rangle\) and \(\mathbf{J}=\langle-1 / \sqrt{2}, 1 / \sqrt{2}\rangle\). Write the vector \langle 2,-6\rangle in terms of \(\mathbf{I}\) and \(\mathbf{J}\).
Carry out the following steps to determine the (smallest) distance between the point \(P\) and the line \(\ell\) through the origin. a. Find any vector \(\mathbf{v}\) in the direction of \(\ell\) b. Find the position vector u corresponding to \(P\). c. Find \(\operatorname{proj}_{\mathbf{v}} \mathbf{u}\). d. Show that \(\mathbf{w}=\mathbf{u}-\) projy \(\mathbf{u}\) is a vector orthogonal to \(\mathbf{v}\) whose length is the distance between \(P\) and the line \(\ell\) e. Find \(\mathbf{w}\) and \(|\mathbf{w}| .\) Explain why \(|\mathbf{w}|\) is the distance between \(P\) and \(\ell\). \(P(-12,4) ; \ell: y=2 x\)
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