Answered: The solution of Laplace's equation Urx… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section: Chapter Questions

Problem 34RE

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Needed to be solve this multiple choice question correctly in 30 minutes and get the thumbs up please show neat and clean work please

The solution of Laplace's equation Ugr + Uny
0 < x < 3, 0 < y< 4, subject to the boundary conditions
u(z, 0)
the form u(x, y) =
O within the rectangular region
wwww
3x – x², u(x, 4) = 0, u(0, y) = 0, u(3, y) = 0. has
an sinh
NT(3-x)
sin ()
NTY
n=1
O b) ∞
E an sinh () sin()
NTY
3
n=1
O c) None of these
O d)
(4-y)
I a, sinh 2 sin(")
3

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Express the given parametric equations of a line using bracket notation and also using i, j, k notation. (a) x = 6t, y = − 4 + 3t O (x, y) = (0, 4) +1 (6,3) ai + bj = -4j + t (6i + 3j) O (x, y) = (0, 4) + t (6, 3) ai + bj = 4j+ t (6i + 3j) O (x, y) = (0, 6) + (-4, 3) ai + bj = −4j + t (6i + 3j) O(x, y) =(4,0) + t (3,6) ai + bj = −4i + t (3i + 6j) O (x, y) = (6,3)+1(0, - 4) ai + bj = (6i + 3j) + (-4j) (b)x= 4 + 6t,y=4+9t, z = 87t O (x, y, z) = (-4, 4, − 8) +1(-6, ai + bj + ck = 9,7) ( − 4i + 4j - 8 k) + t ( − 6i - 9j + 7k) O (x, y, z) = (4, ai + bj + ck = 4, 8) + t (6, 9, - 7) (4i − 4j + 8 k) + t (6i + 9j – 7k) O (x, y, z) = (6,9, −7) +1(4, - 4,8) ai + bj + ck = (6i + 9j − 7k) + t(4i - 4j + 8k) O (x, y, z) = (-6, 9, 7) + 1 - 4, 4, 8) ai + bj + ck = ( − 6i − 9j + 7k) + t( − 4i + 4j − 8 k) O (x, y, z) = (4, 4, 8) + t (6,9,7) ai + bj + ck = (4i +4j + 8 k) + t (6i + 9j + 7k)
A parametric equation of a line in R3 through the point (zo, yo, zo) and parallel to another line with the following parametric equation x = x₁ + at y = y₁ + bt z = 2₁ + ct is 0 x= = (x₁ - x0) + at y = (y₁ - yo) + bt Z= = (2₁-zo) + ct x = x₁ + at y=yo+bt z = 20 + ct tER x = x₁ + (x₁ -xo)t y = yo + (y₁ - yo)t te R z = 20 + (21 - 2o)t tER x= xo + (yoc - zob)t y=yo+ (zoa - xoc)t te R 2= 20 + (zob - yo a)t H= = (xo-x₁) + at y = (30-3₁) + bt z = (20 - 2₁) + ct tER tER
Express the given parametric equations of a line using bracket notation and also using i, j, k notation. (a) x = 6t, y = - 4 + 3t O (x, y) = (0, 4) +1 (6,3) ai + bj = -4j + t (6i + 3j) O (x, y) = (0, 4) + t (6, 3) ai + bj = 4j+ t (6i + 3j) O (x, y) = (0, 6) + (-4, 3) ai + bj = − 4j + t (6i + 3j) O(x, y) =(4,0) + t (3,6) ai + bj = −4i + t (3i + 6j) O (x, y) = (6,3)+1(0, - 4) ai + bj (6i + 3j) +t(-4j) (b)x= 4 + 6t,y=4+9t, z = 87t O (x, y, z) = (-4, 4, − 8) +1(-6, ai + bj + ck = 9,7) ( − 4i + 4j - 8 k) + t ( − 6i - 9j + 7k) O (x, y, z) = (4, ai + bj + ck = 4, 8) + t (6, 9, - 7) (4i − 4j + 8 k) + t (6i + 9j – 7k) O (x, y, z) = (6,9, −7) +1(4, - 4,8) ai + bj + ck = (6i + 9j − 7k) + t(4i - 4j + 8k) O (x, y, z) = (-6, 9, 7) + 1 - 4, 4, 8) ai+bj + ck = ( – 6i – 9j + 7k) + t( − 4i + 4j − 8k) O (x, y, z) = (4, 4, 8) + t (6,9,7) ai + bj + ck = (4i +4j + 8 k) + t (6i + 9j + 7k)
Express the given parametric equations of a line using bracket notation and also using i, j, k notation. (a) x = 61, y = −4+7t O(x, y) = (0, 6) + 1 ( − 4, 7) ai + bj = -4j+ 1 (6i + 7j) O(x, y) = (6, 7) + 1(0, − 4) ai + bj = (6i + 7j) + ¹(-4j) O(x, y) = (-4,0) +1 (7,6) ai + bj = -4i +1 (7i + 6j) O(x, y) = (0, 4) + 1 (6, 7) ai + bj = 4j+1 (6i + 7j) O (x, y) = (0, - 4)+1(6, 7) ai + bj = -4j+1 (6i + 7j)
Draw the contour curves for h (x, y) _y²+x² -1+2x and c = 0, 2, 4.
7
5. Let L₁ be the line with parametric form (x, y, z) = (1, 0, 2) + (-1, 2, 3), and let L₂ be the line with parametric form t ER, (x, y, z)=(3, 4, 16) + u(2, 0, 4), u € R. (a) Find the angle between L₁ and L2 in radians, correct to two decimal places. (b) Let II be the plane containing L₁ and L₂. Find a normal vector for II and the equation of II. (c) Show that there is a unique point where L₁ and L2 intersect, and find its coordi- nates. (d) Find four points A, B, C, D in R³ such that there is no plane containing all of A, B, C and D, justifying your answer.

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