벡터와 공간 전체 문제

For each of the following problems, find the tangential and normal components of acceleration.

$\text{r}(t)=3\cos (2\pi t)\text{i}+3\sin (2\pi t)\text{j}$

Find the position vector-valued function $\text{r}(t),$ given that $\text{a}(t)=\text{i}+{e}^{t}\text{j},$ $\text{v}(0)=2\text{j},$ and $\text{r}(0)=2\text{i}.$

The force on a particle is given by $\text{f}(t)=(\cos t)\text{i}+(\sin t)\text{j}.$ The particle is located at point $(c,0)$ at $t=0.$ The initial velocity of the particle is given by $\text{v}(0)={v}_{0}\text{j}.$ Find the path of the particle of mass m. (Recall, $\text{F}=m\cdot \text{a}.)$

An automobile that weighs 2700 lb makes a turn on a flat road while traveling at 56 ft/sec. If the radius of the turn is 70 ft, what is the required frictional force to keep the car from skidding?

Using Kepler’s laws, it can be shown that ${v}_{0}=\sqrt{\frac{2GM}{{r}_{0}}}$ is the minimum speed needed when $\theta =0$ so that an object will escape from the pull of a central force resulting from mass M. Use this result to find the minimum speed when $\theta =0$ for a space capsule to escape from the gravitational pull of Earth if the probe is at an altitude of 300 km above Earth’s surface.

Find the time in years it takes the dwarf planet Pluto to make one orbit about the Sun given that $a=39.5$ A.U.

Show that the particle moves on a circular cone.

Find the angle between the velocity and acceleration vectors when $t=1.5.$

Find the tangential and normal components of acceleration when $t=1.5.$

A parametric equation that passes through points P and Q can be given by $\text{r}(t)=\langle {t}^{2},3t+1,t-2\rangle ,$ where $P(1,4,-1)$ and $Q(16,11,2).$

Chapter Review Exercises

True or False? Justify your answer with a proof or a counterexample.

$\frac{d}{dt}[\text{u}(t)\,\times \,\text{u}(t)]=2{\text{u}}^{'}(t)\,\times \,\text{u}(t)$

The curvature of a circle of radius $r$ is constant everywhere. Furthermore, the curvature is equal to $1/r.$

The speed of a particle with a position function $\text{r}(t)$ is $({\text{r}}^{'}(t))/(|{\text{r}}^{'}(t)|).$

Find the domains of the vector-valued functions.

$\text{r}(t)=\langle \sin (t),\ln (t),\sqrt{t}\rangle$

Find the domains of the vector-valued functions.

$\text{r}(t)=\langle {e}^{t},\frac{1}{\sqrt{4-t}},\sec (t)\rangle$

Sketch the curves for the following vector equations. Use a calculator if needed.

[T] $\text{r}(t)=\langle {t}^{2},{t}^{3}\rangle$

Sketch the curves for the following vector equations. Use a calculator if needed.

[T] $\text{r}(t)=\langle \sin (20t){e}^{-t},\cos (20t){e}^{-t},{e}^{-t}\rangle$

Find a vector function that describes the following curves.

Intersection of the cylinder ${x}^{2}+{y}^{2}=4$ with the plane $x+z=6$

Find a vector function that describes the following curves.

Intersection of the cone $z=\sqrt{{x}^{2}+{y}^{2}}$ and plane $z=y-4$

Find the derivatives of $\text{u}(t),$ ${\text{u}}^{'}(t),$ ${\text{u}}^{'}(t)\,\times \,\text{u}(t),$ $\text{u}(t)\,\times \,{\text{u}}^{'}(t),$ and $\text{u}(t)\cdot {\text{u}}^{'}(t).$ Find the unit tangent vector.

$\text{u}(t)=\langle {e}^{t},{e}^{-t}\rangle$