Chapter 11: Problem 3
What is the magnitude of the cross product of two parallel vectors?
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An object moves along a path given by $$\mathbf{r}(t)=\langle a \cos t+b \sin t, c \cos t+d \sin t, e \cos t+f \sin t\rangle$$ for \(0 \leq t \leq 2 \pi\) a. Show that the curve described by \(\mathbf{r}\) lies in a plane. b. What conditions on \(a, b, c, d, e,\) and \(f\) guarantee that the curve described by \(\mathbf{r}\) is a circle?
Compute the indefinite integral of the following functions. $$\mathbf{r}(t)=\langle 2 \cos t, 2 \sin 3 t, 4 \cos 8 t\rangle$$
Explain why or why not Determine whether the following statements are true and
give an explanation or counterexample.
a. The vectors \(\mathbf{r}(t)\) and \(\mathbf{r}^{\prime}(t)\) are parallel for
all values of \(t\) in the domain.
b. The curve described by the function \(\mathbf{r}(t)=\left\langle t, t^{2}-2
t, \cos \pi t\right\rangle\)
is smooth, for \(-\infty
Find the function \(\mathbf{r}\) that satisfies the given conditions. $$\mathbf{r}^{\prime}(t)=\langle\sqrt{t}, \cos \pi t, 4 / t\rangle ; \mathbf{r}(1)=\langle 2,3,4\rangle$$
A golfer launches a tee shot down a horizontal fairway; it follows a path given by \(\mathbf{r}(t)=\left\langle a t,(75-0.1 a) t,-5 t^{2}+80 t\right\rangle,\) where \(t \geq 0\) measures time in seconds and \(\mathbf{r}\) has units of feet. The \(y\) -axis points straight down the fairway and the \(z\) -axis points vertically upward. The parameter \(a\) is the slice factor that determines how much the shot deviates from a straight path down the fairway. a. With no slice \((a=0),\) sketch and describe the shot. How far does the ball travel horizontally (the distance between the point the ball leaves the ground and the point where it first strikes the ground)? b. With a slice \((a=0.2),\) sketch and describe the shot. How far does the ball travel horizontally? c. How far does the ball travel horizontally with \(a=2.5 ?\)
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