Problem Set 3 Assigned: Oct. 8 Due: Oct. 29 For exercises 1-3 you may use Matlab. You should show the computation involved, not just give the answer. (Only 0.5 out of 2.5 credit will be given for the correct answer with no computation.) You should be able to do exercise 4 by inspection, certainly without running Matlab. For exercise 4, just give the answer.

Exercise 1 Let P be the plane in R3 containing the three points h1, 0, 1i h1, 2, 3i and h−1, 2, 1i and let Q be the plane containing the three points h5, 2, 1i, h1, 0, 1i, and h−3, −1, −1i. Find the line that is the intersection of P and Q and represent it in parameterized form {p + t · ~v |t ∈ R}

Exercise 2 Find the distance between the line h1, 0, 2i + t · h1, −1, 1i and the line h3, −2, −1i + u · h−2, 1, 1i. Hint: If L and M are two skew lines in R3 (or higher dimension) — that is, two lines that are not parallel but do not intersect — and p and q are their two closest points, then the line containing p and q is perpendicular to both L and M .

Exercise 3 Let P be the same plane as in Exercise 1. Are the two points h12, 16, −2i and h−5, 18, −10i on the same side of P or on opposite sides?

1

Exercise 4 For each matrix M listed below, consider the product M · ~v where ~v is the homogeneous coordinates of a point in two-space. State whether this operation carries out a translation, rigid motion, scale transformation, invertable affine transformation, or degenerate affine transformation; whether or not it leaves the origin fixed; and whether or not it is a reflection. Note that |h0.28, 0.96i| = 1. Given that fact, you should be able to do these by inspection, without putting pencil to paper, let alone running Matlab.

0.28 −0.96 0 0.96 0.28 0 0 0 1

1 0 0

0.28 0.96 −0.96 0.28 0 0

0 3 1 2 0 1

2.8 9.6 0

1 4 0

−9.6 2.8 0

2 3 5 6 0 1

1 3 1

0.96 0.28 0

0 0 1

0 0 1

1 2 4 8 0 0

2

0.96 0.28 0

0.28 0.96 0

0.28 3 −0.96 2 0 1 0 0 1

Exercise 1 Let P be the plane in R3 containing the three points h1, 0, 1i h1, 2, 3i and h−1, 2, 1i and let Q be the plane containing the three points h5, 2, 1i, h1, 0, 1i, and h−3, −1, −1i. Find the line that is the intersection of P and Q and represent it in parameterized form {p + t · ~v |t ∈ R}

Exercise 2 Find the distance between the line h1, 0, 2i + t · h1, −1, 1i and the line h3, −2, −1i + u · h−2, 1, 1i. Hint: If L and M are two skew lines in R3 (or higher dimension) — that is, two lines that are not parallel but do not intersect — and p and q are their two closest points, then the line containing p and q is perpendicular to both L and M .

Exercise 3 Let P be the same plane as in Exercise 1. Are the two points h12, 16, −2i and h−5, 18, −10i on the same side of P or on opposite sides?

1

Exercise 4 For each matrix M listed below, consider the product M · ~v where ~v is the homogeneous coordinates of a point in two-space. State whether this operation carries out a translation, rigid motion, scale transformation, invertable affine transformation, or degenerate affine transformation; whether or not it leaves the origin fixed; and whether or not it is a reflection. Note that |h0.28, 0.96i| = 1. Given that fact, you should be able to do these by inspection, without putting pencil to paper, let alone running Matlab.

0.28 −0.96 0 0.96 0.28 0 0 0 1

1 0 0

0.28 0.96 −0.96 0.28 0 0

0 3 1 2 0 1

2.8 9.6 0

1 4 0

−9.6 2.8 0

2 3 5 6 0 1

1 3 1

0.96 0.28 0

0 0 1

0 0 1

1 2 4 8 0 0

2

0.96 0.28 0

0.28 0.96 0

0.28 3 −0.96 2 0 1 0 0 1