- Verify all the properties 2.3-2.10.
- Compute the following
- 5+ 6
- 5− 6
- −3+
- −3− 2
- −+ 2

- 5
- Find symmetric equations for the line through the points and.
- Find symmetric equations for the line through the points and.
- Symmetric equations for a line are given. Find parametric equations of the line.
- == z + 7
- == z − 7
- = 2 y + 3 = 2z − 1
- == z + 1
- == z + 2
- == z + 1

- Parametric equations for a line are given. Find symmetric equations for the line if possible. If it is not
possible to do it explain why.
- x = 1 + 2t,y = 3 − t,z = 5 + 3t
- x = 1 + t,y = 3 − t,z = 5 − 3t
- x = 1 + 2t,y = 3 + t,z = 5 + 3t
- x = 1 − 2t,y = 1,z = 1 + t
- x = 1 − t,y = 3 + 2t,z = 5 − 3t
- x = t,y = 3 − t,z = 1 + t

- The first point given is a point contained in the line. The second point given is a direction vector for
the line. Find parametric equations for the line, determined by this information.
- ,
- ,
- ,
- ,
- ,
- ,

- Parametric equations for a line are given. Determine a direction vector for this line.
- x = 1 + 2t,y = 3 − t,z = 5 + 3t
- x = 1 + t,y = 3 + 3t,z = 5 − t
- x = 7 + t,y = 3 + 4t,z = 5 − 3t
- x = 2t,y = −3t,z = 3t
- x = 2t,y = 3 + 2t,z = 5 + t
- x = t,y = 3 + 3t,z = 5 + t

- A line contains the given two points. Find parametric equations for this line. Identify the direction
vector.
- ,
- ,
- ,
- ,
- ,
- ,

- Draw a picture of the points in ℝ
^{2}which are determined by the following ordered pairs. - Does it make sense to write +? Explain.
- Draw a picture of the points in ℝ
^{3}which are determined by the following ordered triples. - You are given two points in ℝ
^{3},and. Show the distance from the pointto the first of these points is the same as the distance from this point to the second of the original pair of points. Note that 3 =,4 =. Obtain a theorem which will be valid for general pairs of pointsandand prove your theorem using the distance formula. - A sphere is the set of all points which are at a given distance from a single given point. Find an
equation for the sphere which is the set of all points that are at a distance of 4 from the point in ℝ
^{3}. - A parabola is the set of all points in the plane such that the distance from the pointto a given pointequals the distance fromto a given line. The pointis called the focus and the line is called the directrix. Find the equation of the parabola which results from the line y = l anda given focus with y
_{0}< l. Repeat for y_{0}> l. - A sphere centered at the point ∈ ℝ
^{3}having radius r consists of all pointswhose distance toequals r. Write an equation for this sphere in ℝ^{3}. - Suppose the distance between andwere defined to equal the larger of the two numbersand. Draw a picture of the sphere centered at the pointif this notion of distance is used.
- Repeat the same problem except this time let the distance between the two points be
+.
- If andare two points such that= 1 for i = 1,2, show that in terms of the usual distance,< 1. What would happen if you used the way of measuring distance given in Problem 17 (= maximum of,,.)?
- Give a simple description using the distance formula of the set of points which are at an equal
distance between the two points and.
- Suppose you are given two points andin ℝ
^{2}and a number r > 2a. The set of points described by^{2}+^{2}= 1. This is a nice exercise in messy algebra. - Suppose you are given two points andin ℝ
^{2}and a number r < 2a. The set of points described by^{2}−^{2}= 1. This is a nice exercise in messy algebra. - Let andbe two points in ℝ
^{2}. Give a simple description using the distance formula of the perpendicular bisector of the line segment joining these two points. Thus you want all pointssuch that=. - Show that |αx| =|α||x| whenever x ∈ ℝ
^{n}for any positive integer n.

Download PDFView PDF