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=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:

1 8 4

2 3 3

x y z

3 5 7

2 3 3

x y z

1 8 4

2 3 3

x y z

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)=<3t^2, 6/t, 5>. 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

5 4 4

3 3 3

x y z

5 4 4

5 3 6

x y z

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=-94x-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 ellipsesZ=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)=<cos t, sin t, t>

What is the unit tangent vector to the curve r(t)=<t^2,t-t^2, t^2+3t-2>

at t=2?

1/√(74) <4,-3,7>

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)

2 2 2 2

04 4 sint t dt

Find the length of the curve

from x=1 to x=4.

2 ln( )

2 4

x xf x

7.5 +1/4 ln(4)

Find the curvature of the ellipse <3cos t, sin t> at (3,0)

3

Find the normal vector N(t) for the curve

<4cos t, 4 sin t, t²>

at t=0.

N(0) = 2/√5 <-1,0,1/2>