In given problem, find the equation of the plane having the given normal vector $\mathbf{n}$ and passing through the given point P.

$$\mathbf{n}=2 \mathbf{i}-4 \mathbf{j}+3 \mathbf{k} ; P(1,2,-3)$$

Find the work done by a force $\mathbf{F}=-4 \mathbf{k}$ newtons in moving an object from (0,0,8) to (4,4,0), where distance is in meters.

Demonstrate that the second formula in Theorem A can also be written as $\kappa=\left|y^{\prime \prime} \cos ^3 \phi\right|$, where $\phi$ is the angle of inclination of the tangent line to the graph of y=f(x).

Find the work done by a force $\mathbf{F}=-5 \mathbf{i}+8 \mathbf{j}$ newtons in moving an object 12 meters north.

A car traveling at constant speed v rounds a level curve, which we take to be a circle of radius R. If the car is to avoid sliding outward, the horizontal frictional force F exerted by the road on the tires must at least balance the centrifugal force pulling outward. The force F satisfies $F=\mu m g$, where $\mu$ is the coefficient of friction, m is the mass of the car, and g is the acceleration of gravity. Thus, $\mu m g \geq m v^2 / R$. Show that $v_R$, the speed beyond which skidding will occur, satisfies

$$v_R=\sqrt{\mu g R}$$

and use this to determine $v_R$ for a curve with $R=400$ feet and $\mu=0.4$. Use g=32 feet per second per second.

Find the work done by the force $\mathbf{F}=6 \mathbf{i}+8 \mathbf{j}$ pounds in moving an object from (1,0) to (6,8), where distance is in feet.

Find the work done by a force of 100 newtons acting in the direction $\mathrm{S} 70^{\circ} \mathrm{E}$ in moving an object 30 meters east.

An object moves along the parabola $y=x^2$ with constant speed of 4. Express a at the point $\left(x, x^2\right)$ in terms of $\mathbf{T}$ and $\mathbf{N}$.

Change the following from Cartesian to cylindrical coordinates.

(a) (2,2,3)

(b) $(4 \sqrt{3},-4,6)$

Find the unit vectors perpendicular to the plane determined by the three points (-1,3,0),(5,1,2), and (4,-3,-1).

Let $\mathbf{a}=\langle\sqrt{2}, \sqrt{2}, 0\rangle, \mathbf{b}=\langle 1,-1,1\rangle$, and $\mathbf{c}=\langle-2,2,1\rangle$. Find each of the following:

(a) $\mathbf{a} \cdot \mathbf{c}$

(b) $(\mathbf{a}-\mathbf{c}) \cdot \mathbf{b}$

(c) $\mathbf{a} /\|\mathbf{a}\|$

(d) $(\mathbf{b}-\mathbf{c}) \cdot \mathbf{a}$

(e) $\frac{\mathbf{b} \cdot \mathbf{c}}{\|\mathbf{b}\|\|\mathbf{c}\|}$

(f) $\mathbf{a} \cdot \mathbf{a}-\|\mathbf{a}\|^2$

In given problem, write both the parametric equations and the symmetric equations for the line through the given point parallel to the given vector.

$$(-1,3,-6),\langle-2,0,5\rangle$$

In given problem, sketch the curve over the indicated domain for t. Find $\mathbf{v}, \mathbf{a}, \mathbf{T}$, and $\kappa$ at the point where $t=t_1$.

$$\mathbf{r}(t)=\frac{t^2}{4} \mathbf{i}+2 \cos t \mathbf{j}+2 \sin t \mathbf{k} ; \quad 0 \leq t \leq 4 \pi ; t_1=\pi$$

Respond with true or false to the following assertion. Be prepared to defend your answer.

The vector $\langle 1,-2,3\rangle$ is parallel to the plane $2 x-4 y+6 z=5$.