Jeopardy 203

Formulas 100 Lines 100 Planes 100 Surfaces 100 Curves 100 Formulas 101 Lines 200 Planes 200 Surfaces 200 Curves 200 Formulas 102 Lines 300 Planes 300 Surfaces 300 Curves 300 Formulas 103 Lines 400 Curves 400 Curves 500 Curves 600

The angle between vectors u and v.

Arccos(u·v/|u||v|)

|u x v|=

|u||v|sin(θ), where θ is the angle between the vectors

Volume of parallelopiped determined by vectors u, v, and w.

|u·(v x w)|

Distance from point (x1, y1, z1) to the plane ax+by+cz+d=0Distance from point (x1, y1, z1) to the plane ax+by+cz+d=0.

|ax1+by1+cz1+d| / √(a²+b²+c²)

Symmetric form of line that goes through points (3,5,7) and (1,8,4)

Any of the following:

Equation of line, in parametric form, that goes through point (3,2,1) and is normal to the plane 3x+2z=4y-5

x=3+3t, y=2-4t, z=1+2t

From t=1 to t=3, a particle is moving along the curve r(t)=. At t=3, its velocity becomes constant. What is its position at t=6?

The particle’s position will be: (81,0,5)

What is the line of intersection of the planes 3x+2y+6z = 11 and 4x-2y-3z = 3? Answer in parametric form.

X=2+6t, y=5/2+33t, z = -14t

Find the equation of the plane that goes through the points (6,4,2), (9,7,5), and (11,16,11). Answer in most reduced form.

3x+4y-7z=20

Find the equation of the plane that contains the lines Answer in standard form

9x+y-8z=81

What are the equations of the planes that are parallel to and 3 units away from 4x-4y+2z=9?

4x-4y+2z=-9 4x-4y+2z=27

Find the domain of the function f(x,y)=ln(x²-y²)

The right and left quadrants of the plane. I.e. the right and left regions determined by the lines y=±x

Describe (name the type) of the x=k, y=k, and z=k traces of -x²/25 +y²/36 – z²/4 = 100 And name the type of this surface

X=k traces are hyperbolas centered on y-axis Y=k traces are ellipses Z=k traces are hyperbolas centered on y-axis The surface is a hyperboloid of two sheets

Give the equation, in standard form, of the surface whose projection to the x-y plane are the lines y=±3x/2, and whose projection to the x-z plane is z=9x². What is the name of this surface?

z/36 = x²/4-y²/9 Hyperbolic paraboloid

Give the vector equation of the standard spiral (centered on the z-axis) and with radius 1.

r(t)=

What is the unit tangent vector to the curve r(t)= at t=2?

1/√(74)

Write an integral for the length of the curve <t²,2t-1,cosπ t> From (0,-1,1) to (4,3,1) (no need to evaluate the integral)

Find the length of the curve from x=1 to x=4.

7.5 +1/4 ln(4)

Find the curvature of the ellipse at (3,0)

3

Find the normal vector N(t) for the curve at t=0.

N(0) = 2/√5