13. a. Distance between parallel planes Show that the dis- tance between the parallel planes Ax + By + Cz = D¡ and Ax + By + Cz = D2 is %3D |D - D2| |Ai + Bj + Ck b. Find the distance between the planes 2x + 3y – z = 6 and 2x + 3y – z = 12. c. Find an equation for the plane parallel to the plane 2x – y + 2z = -4 if the point (3, 2, – 1) is equidistant from the two planes. d. Write equations for the planes that lie parallel to and 5 units away from the plane x – 2y + z = 3.

Q: Graph the quadric surfaces given by the equations in Exer- cises 41–48. y2 41. x² = y? x2 42. z2 9.…

A: As per guidelines incase of multiple questions (or) multiple sub-parts asked we are supposed to…

Q: 3-Find the equatium uf tue plone that is parllel and at (& units) Aujay From The plane (x-ay+t=3)

A: The equation of the given plane is  x-2y+z=3

Q: 9. Find the equations of the planes that are parallel to 2.x - 2y + z = 5 such that the distance…

A: We will solve the problem.

Q: Calculate , z² dS if S is the prism whose base is the triangle in the limited xy plane by the x axis…

A: Here given figure is Since prism has 5 surfaces (1)-OAB (2) OADE (3) CDE (4) OBCD (5) ABCE

Q: Find the point closest to origin in R³ that is on both the planes 3x+y+z=5 and x+y+z=1.

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: 2) The distance between the parallel planes ax + by + cz + d1 = 0 and |d1-d2| ax+by+cz + d2 = 0 is…

A: We find required equations of plane parallel to given plane.

Q: 2. Find the distance between the given parallel planes. 2x +y+ 2z = 8 and 4x +2y + 4z +5 = 0

A: # we are entitled to solve one question at a time, please resubmit the other questions if you wish…

Q: Verify Green's theorem for F =, where C is the part of the parabola y = x², and the line y = 4 for…

A: If be a positively oriented,  closed curve  enclosing the region D Then for a vector field F→=P,Q…

Q: a) Show that the distance between the parallel planes Ax + By + Cz = D1 and Ax + By + Cz = D2 is |D1…

A: (a) To show that the distance between two parallel planes Ax+By+Cz=D1 and Ax+By+Cz=D2 is…

Q: 1. Find the distance from the point S(2,-3,4) to the plane x+2y+ 2z =13

A: "Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 4. (a) Show that the line l given by l: 1= 2+3t, y=1+4t, z= 5+ 2t, te R, lies in the plane II given…

A: Given problem:- Show that the line  given by   l : x = 2 + 3t , y = 1 + 4t , z=5+2t ,t in R   lies…

Q: (42) is the distance between the point (1, –3, –2) and the plane x- 4y-3z-2= 0.

A: In this question we have to find the distance between point and plane. Note: As per our policy we…

Q: 4- Find The equation ofthe plane That the origin andat the poiat p(u1)ond at is parallel to theline…

A: Equation of plane

Q: 6. In Section 12.5, you learned that the distance from a point (ro, Yo, zo) to a plane ax+by+cz = d…

A:

Q: A sphere of constant radius r passes through the origin O, and cuts the axes in A, B, C. Prove that…

A: Given: Equation of the sphere with origin at O x2+y2+z2-ax-by-cz=0→1The radius of the sphere is…

Q: 45. Find the distance from the plane x + 2y + 6z = 1 to the plane x + 2y + 6z = 10.

A:

Q: The plane containing the point (2,5,0) and parallel to the plane 3x – 7y + 6z = 8 has equation Ах +…

A: Recall:      The equation of the plane parallel to Ax+By+Cz=D and passes through the point  α,β,γ ,…

Q: A(n)________ is the collection of points in the plane the difference of whose distances from two…

A: The given statement is "A(n)________ is the collection of points in the plane the difference of…

Q: Verify Green's theorem in the plane for M = -2y and N = x , where C is the semi circle x2 + y? = 9,…

A:

Q: A. Show that the planes (x+y- z = 1) and ( 2x - 3y + 4z = 5 ) are neither parallel nor…

A: Let's find.

Q: 1. Consider the planes P and Q given as 2x- 3y +2 = 3 and 4x- 6y + 22 = 5. Are P and Q parallel?…

A:

Q: find the equation of the plane with the given description. Passes through O and is parallel to 4x −…

A: Solution:Given    A plane 4x − 9y + z = 3To find the equation of the plane passess through O and is…

Q: b) If we are at the point (1, 2, 4) on the surface z = xy², and we move a short distance in the…

A: Please check step 2 for solution.!

Q: 7. Find the distance between the point (3, 1, –2) and the plane x + 2y – 2z = 4.

A: Given that plane x+2y-2z=4 and point 3,1,-2To find :Distance between point and plane Solution if…

Q: 7. If 1x, y2 are the coordinates of a point P in the xy-plane,then x is called the__________________…

A: Given that, x,y are the coordinates of a point P in the xy-plane. The objective is to fill in the…

Q: 2. Find the image of the line y = 5x + 7 under collineation a if a((x, y)) = (2у — х, х — 2).

A:

Q: with the guldin's proposition 1 get the contents of the rotating object that occurs if the land…

A: As per the question we are given a 2d region bounded by the curve : y = 6 - 3x - x2 and the straight…

Q: 7. Find the dis to the plane x +2y+ 6z = 10 %3D 8. Find the angles between the planes a+y= 1,…

A: According to guidelines we should have been able to solve only one question at a time #8

Q: 9. Find the angle between the two planes 2x + 3y + z = 3 and x + 3y + 2z = -3

A: Given,           2x+3y+z=3    and   x+3y+2z=-3We know that,           Suppose the two equation be…

Q: Зу — 2z 3D 6

A: Use a graphing utility to plot the plane, 3y-2z=6 .

Q: A plane can be determined by points, that are not on the same line. OA. unable to be determined OB.…

A:

Q: Ex: Find egns of the line thru the pt ( -!,), perpendicular bo the line 3x= 2y == and paralled to…

A:

Q: Find the line of intersection of the planes x- y- 2z 1 and a +y+z=-1. Lütfen birini seçin: a.L : x =…

A: Given equation of plane is x−y−2z=1 and x+y+z=−1. We have to find the line of intersection of the…

Q: A. Show that the planes (x+ y– z = 1 ) and (2x – 3y+ 4z = 5 ) are neither parallel nor…

A: let the normal vectors for the vectors be a1, a2 thus a1=<1,1,-1> and a2=<2,-3,4>. Let…

Q: Evaluate ||x'y dA if D is the triangle in the xy-plane with vertices at (0,0), (5,2), and (0,2).

A:

Q: 8) Find the parametricequation of the Line passes through p(2, 13to 3x+y-22 =5 and perpendicularof…

A:

Q: d the distance between the point (2, -4, 3) and the plane 5x – 2y + z = 10

A:

Q: Find the distance between the planes 4x + 6y – 2z = 4 and –2x – 3y + z = -3.

A: We have to find out the distance between two planes

Q: 9. Suppose that a : R³ → R³ by perpendicular projection onto the plane 3x – 2y + z = 0. Find ↑

A:

Q: 1. Determine the exact value(s) of k for which the planes x– 2y + kz + 7= 0 and x– 2y – 2 = 0 have…

A: Note: According to our answering guidelines we have to answer only one question if multiple…

Q: 7. C is the hexagon in the xy-plane with vertices (5, –5), (5,5), (0, 10), (–5, 5), (-5, –5), and…

A:

Q: 7 Find the Paametric equ. for the line f intersect of the Planes 3x- 6y -2z =15 and 2X+Y-22%3D5?…

A: Given two lines, 3x-6y-2z=15 ....(i) and 2x+y-2z=5....(ii) Putting z=0 in (i) and (ii) we get,…

Q: find the (dis tan ant) between the planes ml and m2 when ml =: 3x - 2y+z = 9 and m2 =: 6x - 4y+ 2z =…

A: Consider the equation of planes m1: A1x+B1y+C1z+D1=0  m2: A2x+B2y+C2z+D2=0Two planes are parallel if…

Q: a) Show that the line I = 0, y = t, z =t where te R a) lies in the plane 6z+4y–4x = 0 b) is parallel…

A: In the question it is asked to solve for the planes having equation given below.

Q: 3. Show that every sphere through the circle x2 +y-2ax+r²=0, z=0 cuts orthogonally every sphere…

A: This is a co-ordinate geometry problem

Q: Verify Green's theorem for F =, where C is the part of the parabola y = x², and the line y = 4 for…

A:

Q: If the line L: X = 3 + 2t , Y= 2t , Z = t intersects the plane X+ 3y - Z =-4 in a point P. The…

A:

Q: 1. Find the Point the Plane X + 3y +27 = 6 that on %3D is nearest to the orig in.

A: To find- Find the point on the plane x + 3y + 2z = 6 that is nearest to the origin.

Q: Find the point of intersection between the planes TT :x+y-z =1, n, : 2x – 3y -9z = 10 and n, : x +…

A: The three planes are, π1=x+y-z=1 π2=2x-3y-9z=10 π3=x+2y-3z=-4

Q: At what point does the line r(t) = (6- 3t)i + (-2 + 2t)j + (-1- 3t)k intersect the yz-plane %3D O a.…

A: