What speed and direction should the jetliner in Example $8$ have in order for the resultant vector to be $500 \mathrm{mph}$ due east?

Find the areas of the triangles whose vertices are given. A(1, 0, 0), B(0, 2, 0), C(0, 0, -1)

The wind passing over a boat's sail exerted a $1000-\mathrm{lb}$ magnitude force $\mathbf{F}$ as shown here. How much work did the wind perform in moving the boat forward $1 \mathrm{mi}$ ? Answer in foot-pounds.

Find the volume of the ellipsoid

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 .$$

Does your formula give the volume of a sphere of radius a if a=b=c?

Express the area A of the cross-section cut from the ellipsoid

$$x^2+\frac{y^2}{4}+\frac{z^2}{9}=1$$

by the plane z=c as a function of c. (The area of an ellipse with semiaxes a and b is $\pi a b$.) Use slices perpendicular to the z-axis to find the volume of the ellipsoid.

The planes 3 x+6 z=1 and 2 x+2 y-z=3 intersect in a line.

Find equations for the line of intersection.

Sketch the surface.

$$x ^ { 2 } + y ^ { 2 } = z$$

The planes 3 x+6 z=1 and 2 x+2 y-z=3 intersect in a line.

Show that the planes are orthogonal.

Express the area A of the cross-section cut from the ellipsoid

$$x^2+\frac{y^2}{4}+\frac{z^2}{9}=1$$

by the plane z=c as a function of c. (The area of an ellipse with semiaxes a and b is $\pi a b$.)

Show that the line in which the planes x + 2y - 2z = 5 and 5x - 2y - z = 0 intersect is parallel to the line x = -3 + 2t, y = 3t, z = 1 + 4t.

How much work does it take to slide a crate 20 m along a loading dock by pulling on it with a 200-N force at an angle of 30° from the horizontal?

Find the areas of the triangles whose vertices are given. A(-6, 0), B(10, -5), C(-2, 4)

Find the distance from the point to the plane. (1, 0, -1), -4x + y + z = 4

An airplane is lying in the direction 25° west of north at 800 km/h. Find the component form of the velocity of the airplane, assuming that the positive x-axis represents due east and the positive y-axis represents due north.