Explain why or why not Determine whether the following statements 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 3 intersect. c. The plane x + y + z = 0 and the line x = t, y = t, 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 ℝ 3 determine a unique plane. g. The vector ⟨−1, −5, 7⟩ is perpendicular to both the line x = 1 + 5 t , y = 3 − t, z = 1 and the line x = 7 t , y = 3, z = 3 + t.
Explain why or why not Determine whether the following statements 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 3 intersect. c. The plane x + y + z = 0 and the line x = t, y = t, 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 ℝ 3 determine a unique plane. g. The vector ⟨−1, −5, 7⟩ is perpendicular to both the line x = 1 + 5 t , y = 3 − t, z = 1 and the line x = 7 t , y = 3, z = 3 + t.
Explain why or why not Determine whether the following statements 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 ℝ3 intersect.
c. The plane x + y + z = 0 and the line x = t, y = t, 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 − t, z = 1 and the line x= 7t, y = 3, z = 3 + t.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Show that the Laplace equation in Cartesian coordinates:
J²u
J²u
+
= 0
მx2 Jy2
can be reduced to the following form in cylindrical polar coordinates:
湯(
ди
1 8²u
+
Or 7,2 მ)2
= 0.
Find integrating factor
Draw the vertical and horizontal asymptotes. Then plot the intercepts (if any), and plot at least one point on each side of each vertical asymptote.
Chapter 13 Solutions
MyLab Math with Pearson eText -- Standalone Access Card -- for Calculus: Early Transcendentals (3rd Edition)
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Introduction to Statistics..What are they? And, How Do I Know Which One to Choose?; Author: The Doctoral Journey;https://www.youtube.com/watch?v=HpyRybBEDQ0;License: Standard YouTube License, CC-BY