2. Find a unitvedctor with positive z-component porpendcular to the plane 2xt2y-2=3 Z-C

Q: Calculate gij for the distance between (x;) = (1, 1, – 1) and (yi) = (0,1,2) in %3D T-2 1] barred…

A: A point in three dimensional coordinate system is given by xi=1,1,-1yi=0,1,2 Now the point in barred…

Q: Find the area of the band cut from the paraboloid x2 + y2 - z = 0 by the planes z = 2 and z = 6.

A:

Q: 10. Suppose that II1 and II2 are parallel planes in R’, given by: Il : ajx + bịy + C1z = di, and II2…

A: Given two plane which are parallel  Π1:a1x+b1y+c1z-d1,  Π2:a2x+b2y+c2z-d2 and another plane is…

Q: Show by Vector mothads that th ditonce hom fhe fornt P.(Xx, Y,, Z) to the plane Ax -+By+cz-D26 t of…

A:

Q: Find a vected that is Perpenditular to the Plane that Contaiins the e poin +s P(2,3,4), Q~2)0,5), R…

A: Given: The three points:  P (1 , 3 , 4 ) , Q ( - 1 , 0 , 5 ) and R ( 4 , 2 , 2 ).   To determine: A…

Q: Find the ar ea of sur face x - 2y - 2z = 0 that lies above the triangle bounded by the lines x = 2,…

A: Given :          The surface x2 - 2y - 2z = 0 that lies above the triangle bounded by the lines x…

Q: Find a parametrization of the line through P = (4, 9, 8) perpendicular to the yz-plane.

A: Equation of Linethe Line through P =(x0,y0,z0) in the diresction of vector v→=(a,b,c)is defined…

Q: State the equstion of the yz plane

A: We have to find equation of yz plane

Q: 2. Show that the xy plane W = (x, y,0) in Rº is generated by: (i)u = and v = (ii)u : and v = 3

A:

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

A: Given : Vertices : (5,-5) , (5,5), (0,10), (-5,5), (-5,-5) and (0,-10) Integral given :…

Q: Compute the distance d from y to the plane in R' spanned by u1 and uz.

A: We have to find the distance d from y to the plane in R^3.

Q: Find the trace of the plane 3x − 2y + 5z = 4 in the xy-plane.

A: The given equation of the plane is 3x-2y+5z=4. The trace of the plane in the xy-plane is the…

Q: If A = 18zî – 12j + 3yk then find SS,A.ñds where S is the surface of the plane %3D 2x + 3y + 6z = 12…

A: Given, surface S: 2x+3y+6z=12           and                       A→=18zi^-12j^+3yk^

Q: a) Evaluate (-ri-yj+2z k) n dS, where a is the portion of the paraboloid : 3x +3y between the planes…

A: We have to Evaluate the Integral using polar coordinate

Q: 5. Evaluate the integra 21-dA, RZ-3y %3D where R is the parallelogram enclosed by the lines 2r-y 0,…

A:

Q: Find the distance between the planes 5x - 3y + z = 9 and 5x 3y + z = -2.

A:

Q: .Calculate the dstance between tve point (3, -2) and the plane Z Xty-2+1=0.

A: The distance between the point (a, b, c) and the plane Ax+By+Cz+D=0 is given by,

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

A:

Q: Find the area of the part of the paraboloid z=x +y² that lies under the plane 2=9.

A:

Q: What is the projection of r(t) = ti + t4j + etk onto the xz-plane?

A: Given: r(t)= ti+t4j+etk

Q: Find vohume fome above ZVy-xLy, below Z0(xy plane) inside xy/ 3.

A:

Q: I, Use Stoke's theorem to evalueate S.Fidr, where F Cau2)= - it +x3 + 2î aind C is the Ceurve of…

A:

Q: The tangent plane to the elliptic paraboloid z = 2x2 + y² at the point (1, 1, 3) is %3D 1) 8x + 6y +…

A:

Q: Show thata perojective plane of gives 0nder indeed BİBD (+n+l, n+),1)

A: We need to show that the projective plane of order n indeed gives a BIBD (n2+n+1,n+1,1)

Q: Find the area of the surface of the part of the plane 3x + 2y + z = 6 that lies in the first octant.

A:

Q: Find the surface area of the part of the plane z = 4+ 7x + 6y that lies above the rectangle [1, 6] ×…

A: We can use integral calculus to find the surface area of the part of the plane above the given…

Q: is ndicular tò the plane containing the points: A = (-1,1, 2) B = (-3, 2, 1) C = (5 – 1, 1) the

A: Given that the plane is passes through the point A, B and C. We have to find the normal vector of…

Q: Find a parametrization for the line segment beginning at P, (-5, - 2,3) and ending at P2 0, 5/4

A: We have to use parameterization of the line.

Q: 2. Use analytic geometry to verify that quadrilateral JKLM is a parallelogran 1C). YA K(-1,1) 2 -2 0…

A:

Q: 2. Show that the xy plane W = (x, y, 0) in R³ is generated by: (i)u = 2 and v= |1| (ii)u and v = 3…

A: Given the plane: W=(x,y,0) in ℝ3.Let (x,y,0) be any element from W. For a and b in ℝ, we…

Q: Find an equation of the plane through the origin that is parallel to the plane 4x – 3y + 5z + 10 =…

A:

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: Find all planes which (i) are tangent to the elliptic paraboloid z = x² + y², and (ii) pass through…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Find the distance between the planes 4x + 5y + z = 25 and 4x + 5Y + z = 53. %3D

A: Explanation of the solution is given below....

Q: Calculate ff A.n ds If A = yi + 2xj – zk and S is a 2x + y = 6 plane surface in the first octane cut…

A:

Q: 3. Find the orthogonal trayectories to the cuve y = ax²- acx Is held fixed. 3 .2 %3D where a

A: The curves that cut every member of a given family of curves at right angles, is called an…

Q: Find the area of on of the triangle that can drawn between the planes 3x – 6y – 2z = 15 - and 2r + y…

A: For any triangle that will lie between two planes, one point has to lie on a plane and the other two…

Q: Find the points on the hyperboloid 9x^2  - 45y^2 + 5z^2 = 45 where the tangent plane is parallel to…

A:

Q: The part of the plane 3x + 2y +: - 6 that lies in the first octant

A:

Q: Find the area of the part of the plane defined by 3z + 2y + z 12 that lies in the first octant.

A: Given equation of plane :  3x+2y+z= 12

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: Find a parametrization for the line segment beginning at P, and ending at P2. P (2,-3,2) and P2(0, –…

A:

Q: a) Evaluate /| (-ri-yj+2: k) · n dS, where o is the portion of the paraboloid z = 3x2 + 3y? between…

A: Define a function fx,y=z or fx,y=3x2+3y2.  So, the flux integral is:…

Q: Find the area of on of the triangle that can drawn between the planes 3x – 6y – 2z = 15 and 2r + y -…

A: The equations of the two planes are: 3x-6y-2z=152x+y-2z=5 As, 5,0,0 satisfies the equation of plane…

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: Show that a indeed perojective plane of gives 0rder BİBD (n+n+,n+),!

A: We need to prove that a projective plane of order n indeed gives a BIBD (n2+n+1, n+1, 1)

Q: Find the surface area of the part of the plane z = 4 + 6x + 2y that lies above the rectangle [1, 4]…

A:

Q: Evaluate JJ (² where o is the first-octant portion of the plane 2x + y + 2z = 6 (x² + 2yz)dS

A:

Q: Find the distance between the planes  5x − 3y + z = 5  and  5x − 3y + z = −9.

A: Two planes  and will always be parallel if The given planes  and  are parallel because we can see…