Determine whether the following vectors are orthogonal (i.e. perpendicular), parallel, or neither.
  • and are .
  • and are .
  • and are .
  • and are .
Which of the following expressions make sense as a mathematical expression and which do not? Here, are all three dimensional vectors, denotes the dot product, and denotes the cross product.

The acute angle between the lines and is .
If and is a vector such that the angle between and is , then .
Let be a cube of side length .
  • The (acute) angle between the main diagonal of the cube and one of its edges is equal to .
  • The angle between the main diagonal and the diagonal of one of its faces is equal to (you can and should leave your answer as of a number): .

The area of the parallelogram with vertices , , , and is equal to .
The area of the triangle with vertices , , and is equal to .
Consider the four points , , , and . The volume of the parallelepiped spanned by , , and is equal to .
  • Are the vectors , , and coplanar (i.e. do they lie on the same plane)?
  • Are the points , , and coplanar?

If and are unit vectors with , then the angle between and is equal to .
True or False (true means always true, otherwise it is false): If and are nonzero, three-dimensional vectors, then must be parallel to .
The statement above is true if the angle between and is .
Consider the line given by and . Are the lines and parallel, intersecting, or skew?
Let be the line through and . At how many points does intersect the -plane? .
An equation of the plane through , , and in the form is
Let be the line given by , and let be the plane through perpendicular to . The point at which intersects is
Consider the planes and .
  • The acute angle between the planes is .
  • Find a vector equation for the line of intersection between the planes.

The distance from the point to the plane given by is equal to .
Are the planes and parallel or intersecting?
For each of the following, give the answer in the form . Find an equation of the plane:
  • Passing through and normal to :
  • Passing through and parallel to :
  • Containing the line , , and parallel to :
  • Passing through and containing the line , , :

Consider the origin and the plane .
  • The distance from to the plane is .
  • The point on which is closest to is .

Parametrize the following curves. Remember to give bounds on the parameter if necessary.
  • The line segment between the points and
  • The intersection of the surface and the plane .
  • The intersection of the cylinder and the plane .
  • The intersection of the surfaces and .
  • The line tangent to the curve at the points where .
  • The intersection of the cylinder and the cone .

The length of the curve from to is .
Suppose , that and . Find .
Consider the curve given by . At how many points is the tangent vector to the curve horizontal? .
List the values in increasing order:
Suppose a particle is moving with position . At what value of is the speed of the particle smallest? .
Consider the curves and . At what point do the curves intersect?
Suppose , where is a differentiable function. If , , and , then
Consider the function .
  • For which values of does the level curve have no points? It is for all .
  • What do the level curves of look like?

An equation for the tangent plane to the surface at the point is (in the form ):
An equation for the tangent plane to the surface given implicitly as at the point is (in the form ):
Suppose . Then at the point is equal to .
For each of the following, calculate the partials , (and where appropriate).
  • .
  • .
  • .
  • (restricted to the domain where ).

If , find . That is, find .
Let be the surface consisting of the set of points equidistant from the point and the plane .
  • An equation for the surface is
  • The surface is a:

Suppose a surface contains the parametric curves and . An equation for the tangent plane to at is (in the form ):
  • The surface given by is a:
  • The surface given by is a:
  • The surface given by is a:

Let .
  • Which of the following describes the level curve ?
  • The partials
  • The equation of the tangent plane to at the point is (in the form ):
  • What is the rate of change of at in the direction of ?
  • The unit vector direction in which increases most rapidly at the point is:

Let be the surface consisting of all points for which the distance between and the -axis is twice the distance from to the -plane. The surface is a:
The equation of the surface is .
The function satisfies which of the following equations (select all that apply).
The function , where is a constant, satisfies which of the following equations (select all that apply).
If and , , then at is equal to .
The temperature at a point is given by , measured in Celsius. A bug crawls so that its position at seconds is and , where and are in centimeters. The temperature satisfies and . How fast is the temperature rising on the bug’s path after seconds?
If and , , and , then at the point where and :
At which points is the direction of fastest increase of the function equal to :
Do the three points , , and lie on one line?
How many lines are there which pass through and are perpendicular to the vector ?
Consider the function . How many critical points does the function have? .
The points happen at and .
The point is a and the point is a .
The absolute maximum of the function on the region is equal to .
Consider . How many critical points does the function have? .
The point happens at .
The point is a .
Let . How many critical points does the function have? .
The points happen at and .
The point is a and the point is a wordChoicelocal maximumlocal minimumsaddle point.
If , find the absolute maximum and minimum of on the rectangle with vertices , , , and .
Consider the function . On the disc , the absolute maximum value is and the absolute minimum value is .