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Chapter 12: Problem 64

Find the points (if they exist) at which the following planes and curves intersect. $$y=1 ; \mathbf{r}(t)=\langle 10 \cos t, 2 \sin t, 1\rangle, \text { for } 0 \leq t \leq 2 \pi$$

Short Answer

Expert verified
Answer: The intersection points are \(\left(5\sqrt{3}, 1, 1\right)\) and \(\left(-5\sqrt{3}, 1, 1\right)\).

Step by step solution

01

Substitute parametric equation components into the plane equation

Take the \(y\)-component of the curve equation (\(2 \sin t\)), and substitute it into the given plane equation: $$1 = 2 \sin t.$$
02

Solve for t

Divide both sides by 2 to isolate \(\sin t\): $$\frac{1}{2} =\sin t.$$ Now, find the values of \(t\) that satisfy this equation: $$t = \arcsin \frac{1}{2}.$$ Here, \(t = \frac{\pi}{6}, \frac{5\pi}{6}\) and \(0 \leq t \leq 2 \pi\).
03

Find the intersection points

Now, plug the obtained values of \(t\) into the parametric equation of the curve to obtain the intersection points: Intersection Point 1: $$\mathbf{r}\left(\frac{\pi}{6}\right) = \left\langle 10 \cos \frac{\pi}{6}, 2 \sin \frac{\pi}{6}, 1\right\rangle = \left\langle 5\sqrt{3}, 1, 1\right\rangle.$$ Intersection Point 2: $$\mathbf{r}\left(\frac{5\pi}{6}\right) = \left\langle 10 \cos \frac{5\pi}{6}, 2 \sin \frac{5\pi}{6}, 1\right\rangle = \left\langle -5\sqrt{3}, 1, 1\right\rangle.$$ Hence, the intersection points are \(\left(5\sqrt{3}, 1, 1\right)\) and \(\left(-5\sqrt{3}, 1, 1\right)\).

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Most popular questions from this chapter

A 500-kg load hangs from three cables of equal length that are anchored at the points \((-2,0,0),(1, \sqrt{3}, 0),\) and \((1,-\sqrt{3}, 0) .\) The load is located at \((0,0,-2 \sqrt{3}) .\) Find the vectors describing the forces on the cables due to the load.

\(\mathbb{R}^{2}\) Consider the vectors \(\mathbf{I}=\langle 1 / \sqrt{2}, 1 / \sqrt{2}\rangle\) and \(\mathbf{J}=\langle-1 / \sqrt{2}, 1 / \sqrt{2}\rangle\). Show that \(\mathbf{I}\) and \(\mathbf{J}\) are orthogonal unit vectors.

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. $$\begin{aligned} &\mathbf{r}(t)=\langle 1+6 t, 3-7 t, 2+t\rangle;\\\ &\mathbf{R}(s)=\langle 10+3 s, 6+s, 14+4 s\rangle \end{aligned}$$

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 domains of the following vector-valued functions. $$\mathbf{r}(t)=\frac{2}{t-1} \mathbf{i}+\frac{3}{t+2} \mathbf{j}$$
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