(b) Let the distance between two adjacent planes parallel to the plane (hkf) in the direct lattice be d. Show that d = , where G = hA + kB + {C.

Q: In AABC, X is the centroid. If CW = 15 , find XW. %3D

A: Given,           In △ABC, X is the centroid and CW¯=15

Q: Determine the Miller indices of a plane that makes intercepts of (4, 3 and 8) on the coordinate axes…

A:

Q: 6. For any closed surface S, prove that || CurlF.N dS=0.

A:

Q: Is the set c~(R* U {0}) “the entire plane without the non-negative reals" conformally equivalent to…

A: We need to evaluate the statement "the entire plane without non negative reals, ℂ-{ℝ∪{0}}, is…

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: 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: 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: Draw an obtuse triangle and, by construction, find the centroid of the triangle. (HINT: Begin by…

A: To get an obtuse triangle, draw a triangle with one of its interior angle should be greater than…

Q: If S is projected onto the x – y plane, what is the shape of the resulting projection?

A: Idea: if we project the surface in xy- plane z=0. if we substitute z=0 in surface equation we get…

Q: Let M be an elliptic paraboloid z = y² + 2?. (A) "Compute the Gauss map and the shape operator. (B)…

A: Answer for sub question a: Given x=y2+z2 A parametrization is given by: Xu,v=u2+v2,u,v Then,…

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: show that the points AC!,l,4) B(2,2,2), coplanear (in the c(o,l,5) and D (1,0,5) are Same plane).

A:

Q: Prove that the planes determined by the equations ax + by + c z = d and αx + β y + γ z = δ are…

A: Given planes, ax + by + c z = d αx + β y + γ z = δ

Q: Determine the Miller indices of a plane that makes intercepts of (4, 3 and 8) on the coordinate axes…

A: To calculate Miller indices we have to take reciprocal of ratio of intercepts and axial length

Q: Find the volume of a parallelepiped whose edges are ä = 7i+5j-k, 6-j+4k and e= 4i-3j+6k %3D

A:

Q: 2. Consider a natural cubic spline model with two knots at ci and c2 given by y = Bo + B1r+ B2(r -…

A: Given, spline model, y=β0+β1x+β2x-c13+β3x-c33+∈                         β2+β3=0  ,  β2c1+β3c2=0…

Q: In AQUW, point T is the centroid, and VT = 5 . Find QT and VQ. W V QT= VQ =

A: We know centroid divide each median in 2:1 ratio.

Q: Show that the rank of the linear mapping 7.32. Bar Y3= 301 +302 y = 201 +20203, 31-02-203 1 V2 is…

A: To prove that the rank of the matrix (representing the given linear transformation ) is 2 and to…

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

A:

Q: In ANPQ, U is the centroid. If NS=15 find US. R S U

A:

Q: | Let C be the positively oriented square with vertices (0, 0), (2, 0), (2, 2). (0, 2). Use Green's…

A: greens theorem

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

A: Let's find.

Q: Find a conformal map that takes {z|0<argz <} onto the unit disk.

A: Given that a conformal map that takes z | 0&lt;argz&lt;π8 onto the unit disk.

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

A: We will find out the required value.

Q: Let M be an elliptic paraboloid a = y? + z². Compute the Gauss map and the shape operator. Find k…

A:

Q: Use Green's theorem to calculate where C is the rectangle with vertices (0, 0), (2, 0), (2, 3), (0,…

A:

Q: evaluate $ A. dš over the sphere R = 3 and show whether Divergence theorem holds or not. %3D

A: Calculate surface integral over R=3 or dr=0 ∮A→×ds→=∮R cos2θ,π,0×r2sinθdθdϕ,rdϕdr,rsinθdrdθ…

Q: Let C denote the triangle with vertices at (0, 0), (0, 1) and (2, 1), oriented positively. Then f(x²…

A: We have to apply the green's theorem here.

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: Evaluate Sf,(z+ 3y – x²) dS where S is the portion of z = 2 – 3y + x² that lies over the triangle in…

A: Ans :

Q: 4. A 5m x 5cm is cut from a corner of 20cm x 30cm. cardboard, find the centroid from the longest…

A:

Q: 4 6 Let y = 7 2 40 Compute the distance d from y to the plane in IR spanned by ui and uz d =

A:

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: In space coordinates, the xy plane consists of all the points of the form O (0, y, z) O (X, y, z) O…

A:

Q: Surface o is a plane a +y + z = 1 that lies in the first octant. Evaluate

A: Given, surface σ is a plane x+y+z=1 that lies in the first octant. To find              ∬σ(2x+z)dS=…

Q: In R, let S be the piece of the plane 3x + 3y + 3z = 3 that lies in the first octant, oriented…

A: Given S is the piece of the plane 3x+3y+3z=3 that lies in the first octant oriented upward. The…

Q: Let S be the triangular prism with the triangular face in the plane y-3 removed. Let C be the…

A: Given curl F→=xi^-yj^ and the given plane is y=3 So, ϕx,y,z=3j^ Therefore, calculate curl F→·n^ curl…

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

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Q: Find a plane containing t and orthogonal to the pla

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Q: Give an example of a parametrisation of the unit circle, C = {(x,y) E R² |x²+y² = 1}, that generates…

A:

Q: Let T be the tetrahedron with vertices (5, 0, 0), (0, 2, 0), (0, 0, 5) and the origin. Write II|…

A: Given T is a tetrahedron with vertices (5,0,0) , (0,2,0), (0,0,5) and  (0,0,0)…

Q: Consider the solid bounded by surfaces S1, S2, S3, S4, S5, and S6 with vertices A, B, C, D, E and O,…

A: In this question, we use the given sketch of the surface and find the parametrization for each…

Q: Let D* be a parallelogram in the uv-plane with vertices A*(o)0), B*(를/듣), C*(를 ') E*( ).Let D be a…

A:

Q: What are the perpendicular generators on (D) 등 =D3 %3D (P) -= of the surface

A: Given equations are  x216-y24=z ......(1)x3=y2=z-4  ......(2) Equation of generators of the…

Q: Consider the case D+ ={(1,0),(0,1)} = {x1,x2} = {e1,e2} D− ={(−1,−1)} = {x3} = {−e1 −e2} D=D−∪D+.…

A: Please solve. MUST  Verify your solution using R or Python. Please include code and printout as part…

Q: Q1- Determine the coordinates of the centroid of the area.: (1, 1)

A: We need to find the x coordinate and y coordinate of the shaded region which is bounded by two…

Q: Let P be the plane 2r + u – 7 = 2 and let 0(1.0.3) be a poir

A:

Q: Evaluate z dv, where E is enclosed by the paraboloid z = x² + y2 and the plane z = 16.

A: