Linear Algebra Topics - Computer Graphics and Geometric Modeling - page 58

1

s

0 0

Ê ˆ

Á
Á
Á

˜
˜
˜

O

+

D = .

0 1 s

r

Ë ¯

0 0

One uses Theorem 1.11.3(2) and shows that VD + U T

Proof. satisfies the appropriate

identities.

1.12 E XERCISES

Section 1.2

1.2.1. Suppose that the equations

ax += by c and a x ¢ +¢ b y =¢ c

define the same line L. Show that a¢=ka, b¢=kb, and c¢=kc for some nonzero real
number k.

1.2.2. Show that the equation form and point-direction-vector form of the definition of a line

in the plane agree.

1.2.3. Find the equation for all lines in the plane through the point (2,3).

1.2.4. Find the parametric equations of the line through the points (0,1,2) and (-1,-1,-1).

If p , q ΠR n , show that [ p , q ] = [ q , p ]. 1.2.5.

1.2.6. Let a, b Œ R with a £ b. Show that the interval [a,b] consists of the same numbers as

the segment [a,b] where a and b are thought of as vectors. The difference between a
segment and an interval in R is that the interval [a,b] is defined to be empty if b < a,
whereas this is not the case for segments. In fact, as segments (in R 1 ) [a,b] = [b,a].

1.2.7. Consider the line L through (1,-1,0) with direction vector (-1,-1,2). Find the two points

on L that are a distance 2 from the point (0,-2,2).

Section 1.3

1.3.1. Find the cosines of the angles between the following pairs of vectors. Which pairs are

perpendicular? Which pairs are parallel?
(a) (3,1), (1,3) (b) (1,2), (-4,2) (c) (1,2), (-4,-8) (d) (-3,0), (2,1)

Section 1.4

1.4.1. Fill in the missing details in the proof of Theorem 1.4.4.

1.4.2. Prove Theorem 1.4.6.

1.4.3. Find the orthogonal projection of (-1,2,3) on (1,0,1).