4. a. Find the solution of the one-dimensional wave equation J²u მ2 J²u მე:2 by introducing a change of variables from (x, t) to two moving coordinates (ε,n), one moving to the left (with velocity -c) and one moving the right (with velocity c): xct and n = x+ct b. Show that c² in the one-dimensional equation has the dimensions of velocity squared.

Find both the vector equation and the parametric equations of the line through (6,5,4) that is perpendicular to the lines x=8-4t, y = 7+ 7t, z = 9-6t and x = -4t, y = 7+t, z= 9-t, where t = 0 corresponds to the given point.

Suppose point A(9,10) was rotated 270 degrees counterclockwise around the origin.  What is the y coordinate of A'?

Suppose the straight lines L₁ and L2 have respective vector equations 5 --0--0-0) and 8 = where s and t are scalar parameters. 4 -3 + t = 9 -5 2 b) Find the exact value of the cosine of the acute angle 0, between L₁ and L2. cos Ꮎ =

What is the vector parametric equation L(t) for the line through the points (5,-5,0) and (0,-3,3)?

Take the parametric equations x/12 = y/4 = z/9 and x-12/-10 = y-4/-4 = z-9/-7 find the scalar equation of the lines in the form of ax+by+cz+d=0

5. Given the points P=(2,–4,-1) and Q=(5,-4,2) in space. A) Find the parametric equations of the line through points P and Q. B) Express the equation of the line above as a vector valued function, r(t), using proper vector valued function notation.

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…

What is the vector equation of the line segment that connects point P(1,−1,2) to point Q (4,1,7)?

find the vector equation for the line through the point A = (5, -2, 1) at t = 0. and B = (-3, 1, -2) at t = 1.