These are meant to be questions to help you gauge where you are in terms of your level of understanding of the material so far. This is the official half way point between the start of the semester and the first midterm, so I thought this would be a good time for it. They are split up into definitions/general philosophy, quick answer problems, medium questions, and then harder problems.

Some problems have hints.

Definitions and General Philosophy

A unit vector is a vector that has length .
Which of the following is true of , the projection of onto ? Select all that apply.
To define a line in any dimension, we need a point and a:
To define a plane , we need a point and a:
The normalization of a nonzero vector is a vector of length which is direction of the vector .

Problems

Let be the vector . Then:
  • .
  • .
  • .
  • .
  • .
  • The normalization .

Suppose . Then the tangent vector to at the point is
The equation of the plane through the point and perpendicular to the vector is (give the answer in the form ):
A vector equation for the line through the points and is given by
If and , then
An equation for the plane through the point which is perpendicular to the line given by symmetric equations
is (give the answer in the form ):
Are the lines and parallel, skew, or intersecting?
An equation for the plane which contains both the point and the line given by is given by (give the answer in the form ):
The volume of the parallelepiped formed by the vectors , , and is equal to .
The arc length of the portion of the curve which is above the -plane and below the paraboloid is equal to .
Find an equation of a plane whose intersection with the plane is the line . (Nothing to input here, see the hint if you want a possible solution.)
The point on the plane closest to the point is .

True/False and Multiple Choice

The graph of is a:
The level curves of the function are:
True/False: For a function , if , then the two level curves and can intersect.
True/False: The map for parametrizes the intersection of with the plane .
True/False: The integral gives the arc length of the portion of the curve for .
True/False: If and are nonzero vectors with and , then it must be true that .
Which of the following vector valued functions parametrize the curve for . Select all that apply.