벡터와 공간 전체 문제

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

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

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

Find the tangential and normal acceleration components with the position vector $\text{r}(t)=\langle \cos t,\sin t,{e}^{t}\rangle .$

A Ferris wheel car is moving at a constant speed $v$ and has a constant radius $r.$ Find the tangential and normal acceleration of the Ferris wheel car.

The position of a particle is given by $\text{r}(t)=\langle {t}^{2},\ln (t),\sin (\pi t)\rangle ,$ where $t$ is measured in seconds and $\text{r}$ is measured in meters. Find the velocity, acceleration, and speed functions. What are the position, velocity, speed, and acceleration of the particle at 1 sec?

The following problems consider launching a cannonball out of a cannon. The cannonball is shot out of the cannon with an angle $\theta$ and initial velocity ${\text{v}}_{0}.$ The only force acting on the cannonball is gravity, so we begin with a constant acceleration $\text{a}(t)=-g\text{j}.$

Find the velocity vector function $\text{v}(t).$

Find the unit vector in the second quadrant of $\mathbb{R}^2$ that makes a $30^\circ$ angle with the $y$-axis.

In Exercises 23–26, find the area of the parallelogram defined by the given vectors.

$\vec u = \left\langle 1,1,2\right\rangle$, $\vec v = \left\langle 2,0,3\right\rangle$

In Exercises 23–26, find the area of the parallelogram defined by the given vectors.

$\vec u = \left\langle -2,1,5\right\rangle$, $\vec v = \left\langle -1,3,1\right\rangle$

In Exercises 23–26, find the area of the parallelogram defined by the given vectors.

$\vec u = \left\langle 1,2\right\rangle$, $\vec v = \left\langle 2,1\right\rangle$

In Exercises 23–26, find the area of the parallelogram defined by the given vectors.

$\vec u = \left\langle 2,0\right\rangle$, $\vec v = \left\langle 0,3\right\rangle$

In Exercises 27–30, find the area of the triangle with the given vertices.

Vertices: $(0,0,0)$, $(1,3,-1)$ and $(2,1,1)$.

In Exercises 27–30, find the area of the triangle with the given vertices.

Vertices: $(5,2,-1)$, $(3,6,2)$ and $(1,0,4)$.

In Exercises 27–30, find the area of the triangle with the given vertices.

Vertices: $(1,1)$, $(1,3)$ and $(2,2)$.

In Exercises 27–30, find the area of the triangle with the given vertices.

Vertices: $(3,1)$, $(1,2)$ and $(4,3)$.

In Exercises 31–32, find the area of the quadrilateral with the given vertices. (Hint: break the quadrilateral into 2 triangles.)

Vertices: $(0,0)$, $(1,2)$, $(3,0)$ and $(4,3)$.

In Exercises 31–32, find the area of the quadrilateral with the given vertices. (Hint: break the quadrilateral into 2 triangles.)

Vertices: $(0,0,0)$, $(2,1,1)$, $(-1,2,-8)$ and $(1,-1,5)$.

In Exercises 33–34, find the volume of the parallelepiped defined by the given vectors.

$\vec u = \left\langle 1,1,1\right\rangle$, $\vec v=\left\langle 1,2,3\right\rangle$, $\vec w = \left\langle 1,0,1\right\rangle$