An electromagnetic standing wave in a certain material has frequency $1.20 \times 10^{10} \mathrm{~Hz}$ and speed of propagation $2.10 \times$ $10^8 \mathrm{~m} / \mathrm{s}$. $(a)$ What is the distance between a nodal plane of $\overrightarrow{\boldsymbol{B}}$ and the closest antinodal plane of $\overrightarrow{\boldsymbol{B}}$ ? $(b)$ What is the distance between an antinodal plane of $\overrightarrow{\boldsymbol{E}}$ and the closest antinodal plane of $\overrightarrow{\boldsymbol{B}}$ ? $(c)$ What is the distance between a nodal plane of $\vec{E}$ and the closest nodal plane of $\overrightarrow{\boldsymbol{B}}$ ?

Show that the line $x=0, y=t, z=t$ (a) lies in the plane $6 x+4 y-4 z=0$; (b) is parallel to and below the plane $5 x-3 y+3 z=1$; (c) is parallel to and below the plane $6 x+2 y-2 z=-3$.

A loaded penguin sled weighing 80 N rests on a plane inclined at angle $\theta=20^{\circ}$ to the horizontal. Between the sled and the plane, the coefficient of static friction is $0.25$, and the coefficient of kinetic friction is $0.15$. What is the least magnitude of the force $\vec{F}$, parallel to the plane, that will prevent the sled from slipping down the plane?

Find a point on the surface z=

$$3 x ^ { 2 } - y ^ { 2 }$$

at which the tangent plane is parallel to the plane 6x+4y-z=5.

A plane trip lasts 8.0 h; the average speed of the plane during the flight relative to Earth is 220 m/s. What the time difference between an atomic clock on board the plane and one on the ground, assuming they were synchronized before the flight? (Ignore general relativistic complications due to gravity and the acceleration of the plane.)

Uri is on a flight from Boston to Los Angeles. His plane is traveling $20^{\circ}$ south of west at $500 \mathrm{mph}$. Val is on a flight from Miami to Seattle. Her plane is traveling $30^{\circ}$ north of west at $500 \mathrm{mph}$. Somewhere over Kansas, Uri's plane passes $1000 \mathrm{ft}$ directly over Val's plane. Uri is sitting on the right side and can see Val's plane below him after they pass. Uri notices that the fuselage of Val's plane doesn't point in the direction that her plane is moving. What is the angle between the fuselage and the direction of motion?

Find equations for the plane. The plane through (1, 1, -1), (2, 0, 2), and (0, -2, 1)

Sketch a diagram of the description. $\overrightarrow{A B}$ in plane $U$ intersecting $\overrightarrow{C D}$ at point $E$, and point $C$ not on plane $U$

Draw and describe a cross section formed by a plane intersecting a cube as follows. The plane contains the red edges of the cube.

Find an equation of the plane. The plane through (3, -1, 1), (4, 0, 2), and (6, 3, 1)

An inclined plane, fixed to the inside of an elevator, makes a $32^{\circ}$ angle with the floor. A mass $m$ slides on the plane without friction. What is its acceleration relative to the plane if the elevator accelerates downward at $0.50 \mathrm{~g}$.

A plane is cruising at an altitude of $24,000$ feet. It begins to descend at a constant rate of $20$ feet per second. (a) Write an expression for the altitude of the plane after $t$ seconds. (b) What is the altitude of the plane after $5$ minute?

A small plane flies in a sinusoidal pattern at a constant altitude over a river. The path of the plane is y = 0.1sin(30x), where $0 \leq x \leq 10$ for x and y in km. Find the total distance traveled by the plane during the flight.

Graph on a plane.

$$x>-2$$