Explain why or why not Determine whether the following state- ments are true and give an explanation or counterexample. a. The line r = (3, -1,4) + t(6, –2, 8) passes through the origin. b. Any two nonparallel lines in R° intersect. c. The plane x + y + z = 0 and the line x = 1, y = 1, z = t are parallel. d. The vector equations r = (1, 2, 3) + t(1, 1, 1) and R = (1, 2, 3) + t(-2, –2, –2) describe the same line. e. The equations x + y – z = 1 and -x – y + z = 1 describe the same plane. f. Any two distinct lines in R³ determine a unique plane. g. The vector (-1, –5, 7) is perpendicular to both the line x = 1 + 5t, y = 3 – 1, z = 1 and the line x = 7t, y = 3, z = 3 + t. %3D
Explain why or why not Determine whether the following state- ments are true and give an explanation or counterexample. a. The line r = (3, -1,4) + t(6, –2, 8) passes through the origin. b. Any two nonparallel lines in R° intersect. c. The plane x + y + z = 0 and the line x = 1, y = 1, z = t are parallel. d. The vector equations r = (1, 2, 3) + t(1, 1, 1) and R = (1, 2, 3) + t(-2, –2, –2) describe the same line. e. The equations x + y – z = 1 and -x – y + z = 1 describe the same plane. f. Any two distinct lines in R³ determine a unique plane. g. The vector (-1, –5, 7) is perpendicular to both the line x = 1 + 5t, y = 3 – 1, z = 1 and the line x = 7t, y = 3, z = 3 + t. %3D
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:Explain why or why not Determine whether the following state-
ments are true and give an explanation or counterexample.
a. The line r = (3, -1,4) + t(6, –2, 8) passes through the
origin.
b. Any two nonparallel lines in R° intersect.
c. The plane x + y + z = 0 and the line x = 1, y = 1, z = t are
parallel.
d. The vector equations r = (1, 2, 3) + t(1, 1, 1) and
R = (1, 2, 3) + t(-2, –2, –2) describe the same line.
e. The equations x + y – z = 1 and -x – y + z = 1 describe
the same plane.
f. Any two distinct lines in R³ determine a unique plane.
g. The vector (-1, –5, 7) is perpendicular to both the line
x = 1 + 5t, y = 3 – 1, z = 1 and the line x = 7t, y = 3,
z = 3 + t.
%3D
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