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.
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): .
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?
True or False (true means always true, otherwise it is false): If and are
nonzero, three-dimensional vectors, then must be parallel to .
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.
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 .
Suppose a particle is moving with position . At what value of is the speed of
the particle smallest? .
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 given implicitly
as
at the point is (in the form ):
For each of the following, calculate the partials , (and where appropriate).
- .
- .
- .
- (restricted to the domain where ).
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 ):
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 function , where is a constant, satisfies which of the following equations
(select all that apply).
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?