Find a normal vector of the surface at the given point P.

ax+by+cz=d, any P

Find the shortest distance between the following pairs of nonparallel lines and find the points on the lines that are closest together. (a)

$$\left[ \begin{array} { l } { x } \\ { y } \\ { z } \end{array} \right] = \left[ \begin{array} { l } { 3 } \\ { 0 } \\ { 1 } \end{array} \right] + s \left[ \begin{array} { r } { 2 } \\ { 1 } \\ { - 3 } \end{array} \right] ; \quad \left[ \begin{array} { l } { x } \\ { y } \\ { z } \end{array} \right] = \left[ \begin{array} { r } { 1 } \\ { 1 } \\ { - 1 } \end{array} \right] + t \left[ \begin{array} { l } { 1 } \\ { 0 } \\ { 1 } \end{array} \right]$$

, (b)

$$\left[ \begin{array} { l } { x } \\ { y } \\ { z } \end{array} \right] = \left[ \begin{array} { r } { 1 } \\ { - 1 } \\ { z } \end{array} \right] + s \left[ \begin{array} { l } { 1 } \\ { 1 } \\ { 1 } \end{array} \right] ; \quad \left[ \begin{array} { l } { x } \\ { y } \\ { z } \end{array} \right] = \left[ \begin{array} { r } { 2 } \\ { - 1 } \\ { 3 } \end{array} \right] + t \left[ \begin{array} { l } { 3 } \\ { 1 } \\ { 0 } \end{array} \right]$$

, (c)

$$\left[ \begin{array} { l } { x } \\ { y } \\ { z } \end{array} \right] = \left[ \begin{array} { r } { 3 } \\ { 1 } \\ { - 1 } \end{array} \right] + s \left[ \begin{array} { r } { 1 } \\ { y } \\ { z } \end{array} \right] = \left[ \begin{array} { l } { 1 } \\ { 2 } \\ { 0 } \end{array} \right] + t \left[ \begin{array} { l } { 1 } \\ { 0 } \\ { 2 } \end{array} \right]$$

, (d)

$$\left[ \begin{array} { l } { x } \\ { y } \\ { z } \end{array} \right] = \left[ \begin{array} { l } { 1 } \\ { 2 } \\ { 3 } \end{array} \right] + s \left[ \begin{array} { r } { 2 } \\ { 0 } \\ { - 1 } \end{array} \right] ; \quad \left[ \begin{array} { l } { x } \\ { y } \\ { z } \end{array} \right] = \left[ \begin{array} { l } { 3 } \\ { - 1 } \\ { 0 } \end{array} \right] + t \left[ \begin{array} { l } { 1 } \\ { 1 } \\ { 0 } \end{array} \right]$$

Experiments show that in a temperature field, heat flows in the direction of maximum decrease of temperature T. Find this direction in general and at the given point P. Sketch that direction at P as an arrow. T=x²+y²+4z², P:(2, -1, 2)

The formula a=Vf-Vi divided by t is used to find an object's acceleration given initial velocity Vi, and time t. Solve this formula for Vf

Given the velocity potential f of a flow, find the velocity v=∇f of the field and its value v(P) at P. Sketch v(P) and the curve f=const passing through P. f=x²-6x-y², P:(-1,5)

In each case, find the shortest distance from the point P to the plane and find the point Q on the plane closest to P. (a) P(2, 3, 0); plane with equation 5x+y=z=1. (b) P(3, 1, -1); plane with equation 2x+y-z=6.

Points P, Q, and R are given. a. Find the general equation of the plane passing through P, Q, and R. b. Write the vector equation $\mathbf { n } \cdot \vec { P S } = 0$ of the plane at a., where S(x, y, z) is an arbitrary point of the plane. c. Find parametric equations of the line passing through the origin that is perpendicular to the plane passing through P, Q, and R. P(-2, 1, 4) Q(3, 1, 3), and R(-2, 1, 0)

Find an equation of the plane passing through the point perpendicular to the given vector or line. Point: (3, 2, 2) Perpendicular to: x-1 / 4 = y + 2 = z+3 / -3

The force in an electrostatic field given by f(x, y, z) has the direction of the gradient. Find ∇f and its value at P. f=ln(x²+y²), P:(8,6)

The force in an electrostatic field given by f(x, y, z) has the direction of the gradient. Find ∇f and its value at P. f=x/(x²+y²), P:(1,1)

The force in an electrostatic field given by f(x, y, z) has the direction of the gradient. Find ∇f and its value at P.

Prove and illustrate by an example. ∇(f/g)=(1/g²)g(∇f-f∇g)

Prove and illustrate by an example.

$$∇(fg)=f∇g+g∇f$$

Find $\nabla f$. Graph some level curves $f=$ const. Indicate $\nabla f$ by arrows at some points of these curves. $f=x^2+\frac{1}{9} y^2$