Find a vector parametric equation r(t) for the linethrough the points P = (4, −3, 4) and Q = (3, −6, 2)for each of the given conditions on the parameter t.(a) If 7(0) = (4, −3, 4) and 7(2) = (3, −6, 2), thenr(t) =(b) If 7(6) = P and 7(8) = Q, thenr(t) ==(c) If the points P and Q correspond to the parametervalues t = 0 and t = -4, respectively, thenF(t) =

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Answered: Find a vector parametric equation r(t)…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Find the maximum and minimum values of the function f(x, y) = 2x² + 3y² - 4x-5 on the domain 2 + y² 256. The maximum value of f(x,y) is: List the point(s) where the function attains its maximum as an ordered pair, such as (-6,3), or a list of ordered pairs if there is more than one point, such as (1,3), (-4,7). The minimum value of f(x, y) is: List points where the function attains its minimum as an ordered pair, such as (-6,3), or a list of ordered pairs if there is more than one point, such as (1,3), (-4,7).
(a) Find an equation of the line perpendicular to the vector i + j, and in the plane (x − 9) − (y − 23) + (z − 2020) = 0. (b) Find the distance between this line and the point Q(9, 23, 2019).
the space curve (t) = (2t, t², 2t)
Solve the following equations show your solutions. 1. Given the line 2x = 5y + 9. Find its equation in intercept/s form. 2. Determine the divergence of the vector: V = i(x^2)y + j(-xy) + k(xyz) at coordinates (3, 2, 1). 3. If "n" is any positive integer, then (n - 1)(n - 2)(n - 3)...(3)(2)(1) = ?
16. Find both the vector equation and the parametric equations of the line through (-2,2,8) and (1, - 4,0), where t =0 corresponds to the first given point. The vector equation is (x,y,z) = + (1). Find the parametric equations of the line through (-2,2,8) in the direction from (-2,2,8) toward (1,-4,0). The parametric equations are x = (Use the answer from the previous step to find this answer.) y%3D (1) O t(3,-6,-8). O t(-6,3,-8). O t(-8,-6,3). O t(-8,3,-6). 17. If possible, find the absolute maximum and minimum values of the following function on the region R. f(x,y) = 5 e-X-Y; R={(x,y): x20, y2 0} Determine the absolute maximum value of f(x,y) on R. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The absolute maximum value of f on R is O B. The function f(x,y) has no absolute maximum value on R. Determine the absolute minimum value of f(x,y) on R. Select the correct choice below and, if necessary, fill in the answer box to…
22. Determine the vector and parametric equations for the line passing through the points P(-1,-3) and Q(3,5). a. 7 = (1,-3) + s(4,8), S € R r = (3,5) + s(1,2), s ER x=21-1, y = 4t-3,1 € R x=4t-1, y = 8t-3,1 € R b. 7=(1,0) + s(2,4), S ER d. all of the above x=1-1, y = 2t-3,1 € R x-1 y+1 23. What value of k will make the line with symmetric equations- 2-3 be the same as the line 2 k-1 with vector equation r = (1,-1,3)+s(3,2,2), s € R? a. 3 c. 2 b. 4 d. none of the above =
The equations of lines L₁ and L2 are -(1) + () and r - (-) + ² (1) +t2 r= · and p are parameters (scalars). respectively, where t (i) Show that contains the point (2,-1,-4) but L₂ does not pass through this point; (ii) Find the acute angle between L₂ and the line joining (1,-1,1) and (2,-1,-4).
Take the parametric equations of the lines [x/13 = y/3 = z/8] and [x-11/-10 = y-3/-4 = z-8/-7]; Find the scalar equation of the lines in the form of ax+by+cz+d=0 Show how to check that (13,3,8) fits the equation of the plane. Show how to check how (0,0,0) fits the equation of the plane. Show to to check that (1,-1,1) fits the equation of the plane.
(1) Find the normal vectors of the lines joining the points with the following position vectors (a) 4ị + 7j, 6ị – 3j (b) –2į – 3j, -8į – 7j (c) 2i + j/2, 6+ į/6 (2) Write down in normal form the equation of the line which is (a) parallel to the line r · (5i +3j) = 2 and passes through the point (2, –3); (b) perpendicular to the line 7x + 6y = 1 and passes through the point (1, 1); (c) passes through the points (3, 1) and (–2, –7); (3) Find the equation of the perpendicular bisector of the line join- ing the points with position vectors (a) 3i + 4j, 7i + 1oj (b) –4į – 3j, 2i – 6j. (4) In each of the following cases write down the normal vectors of the given lines and hence find the acute angles between them: (a) 2r – 5y – 3 = 0, 3x+ 7y = 10; (b) 2r + 3y – 2 = 0, 2r – 8y + 19 = 0, (c) 4y – 7y + 13 = 0, 2x – 8y + 19 = 0 (5) Use vector methods to find the position vectors of the points of intersection of the following pairs of lines: (a) r (i – j) +2 = 0, r·(3i+2j)+6 = 0; (b) r· (4i – j) –…

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