Sketch the following Cartesian Products on the plane R? or in space R. a. (-1,0,2} x {1, 2, 3} b. (1,1.5, 2} × [1,2] {(x, y) E R?:x² + y's 1} x [0,1] C.

Q: Which of the following is the equation of the plane that is tangent to the surface 2 x² + y? – x z –…

A: Consider the provided information, The provided surface, 2x2+y2-xz-3z=1 Find the partial derivative…

Q: Find a function r(t) that describes the line through P(3,4,5) that is orthogonal to the plane 2x-z=4

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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: Let P be the tangent plane to the surface z=2x2+y2 at the point Q=(-1,-1,3). Which of the following…

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Q: Let r(t) be a positively oriented rectangle with corners (0,0), (0,1), (2,1) and (2,0) and F(r, y)…

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Q: F = (x² – y², y² – 2², z² – x²), S is the part of the plane r + y + z = 1 in the first octant.

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

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Q: Given, (2z – z) dz along the directed line segment from -i to 1. What is a parametric form of z(t)?…

A: We have to find the parametric form of directed line segment from -i  to 1

Q: Evaluate fs F dS, where F(r, y, z) = (2², xy, z) and S is the part of the paraboloid z = 22 + y?…

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Q: Given F(x, y, z) = -yzi + xzj – xyzk be a vector function that passes through a surface S of z + x2…

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Q: Is v = 2i - 4j + k related in any special way to the plane 2x + y = 5? Give reasons for your answer.

A: Given:

Q: B. Sketch these Cartesian products on the x-y plane R² (or R for the last two). 9. {1,2,3} x…

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Q: Given, (2z – z) dz along the directed line segment from - i to 1. What is a parametric form of z(t)?…

A: We have to form a line segment from -i to 1 Let 0≤t≤1

Q: Let T be the plane x – y – z = 7. Which of the following planes are orthogonal to T and pass through…

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Q: et the affine map A : R? → R² be defined by A(x, y) (3x + 4, 2y – 3). Show that the image of a unit…

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Q: Let P be the tangent plane to the surface z=2x2+y2 at the point Q=(-1,-1,3). Which of the following…

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Q: Show that the curve a(t) = (t,4,²) lies in a plane and find the equation of this plane.

A: Solution:- αt=t,1+tt,1-t2t t,1t+1,1t-t=x,y,z It is clear that, t-1t-1+1t-t=x-y+z x-y+z=-1

Q: Determine the projection of the following surface on the xz plane. (xz trace) 4x – + = 0

A: Given equation of surface is 4x-y24+z28=0

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

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Q: Let z be a complex mumber , f() b any funcion, and c be contour any then to f B(Z)dZ?

A: Let z be a complex number, f(z) be any function and c be any contour.     A contour is a type of…

Q: Which of the following is the equation of the plane that is tangent to the surface 2 x2 + y? x z –…

A: Given: 2x2+y2-xz-3z=1

Q: Let r() = (In1, 1, 5). Find the equation of the osculating plane corresponding to1 = 1.

A: Find the point at t=1 the vector r(t)=lnt,t,t22 corresponds. lnt,t,t22=ln1,1,122=0,1,12=0,1,0.5…

Q: Let z E C. Draw the following regions on the xy-plane. Justify your work. (a) {z E C : 4 < |z| < 6}…

A: Given: a) z∈ℂ: 4&lt;z&lt;6 b) z∈ℂ: z-22=4

Q: Use Stoke's theorem to evaluate F dr where F(x,y, z)=e-7i+ +e"k and C is the boundary of the part of…

A: Let Fx,y,z be a vector field and the surface S is bounded by a simple closed curve C. The Stoke's…

Q: Prove that the locus of the foot of the perpendicular drawn x2 +² a? = 1 to any of its tangent om…

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Q: Let P be the tangent plane to the surface z-2x2+y2 at the point Q=(1,1,3). Which of the following…

A: Tangent plane to given surface

Q: Find all planes in R3 whose intersection with the xy-plane is the liner( t) = t ( 2, 1, 0) .

A: GIVEN  Intersection with the xy-plane is the line r( t) = t ( 2, 1, 0)

Q: Evaluate F.ndS for the given F and o. (b) F(x, y, z) = (x² + y) i+ xyj– (2xz + y) k, o: the surface…

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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 the barycentric coordinates of p with respect to the affinely independent set of points that…

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Q: Sketch the following Cartesian Products on the plane R? or in space IR. a. (-1,0, 2) x (1,2, 3) b.…

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Q: . Evaluate || (2- 1) do, where S is the part of the cone z = Vr + y² which lies under the plane z =…

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Q: Sketch the following Cartesian Products on the plane R? or in space {(x, y) € R²:x² + y² s 1} ×…

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Q: V9/ Compute I ,cxyz), uhere s is the part f the plane +x that lies inside the half - cyllnder…

A: Given: The Integral is ∬SxyzdS. The surface is the part of the plane S  z=y+x that is inside the…

Q: Evaluate x2 zdS, where S is the part of the S plane z = 1+ 2x + 3y above the rectangle [0, 2] × [0,…

A: We have to evaluate ∬Sx2zdS, where S is the part of the plane z=1+2x+3y above the rectangle 0,2×0,3.…

Q: Which of the following planes are not parallel to the plane -x+ 3y + z = 3? Select all that apply. O…

A: Our objective is to find the required plane.

Q: Calculate F- dS if F(x, y, z) = x²ye²i+ æy?j – 2xye*k, where S is the surface of the box bounded by…

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Q: Find an equation for the plane passing through the origin that is parallel to the tan- gent plane to…

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Q: Let P be the tangent plane to the surface z=2x2+y2 at the point Q=(1,1,3). Which of the following…

A: tangent planes to a surface are planes that just touch the surface at the point and are “parallel”…

Q: Describe and sketch the surfaces in space (R') defined by the following equa- tions: (a) y = -z +1…

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Q: Which of the following planes are not parallel to the plane –x + 3y + z = 3? Select all that apply.…

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Q: where F(x, y, z) = (ry, x²y, y) Use Gauss's theorem to calculate where is the paraboloid z = 4 – x²…

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

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Q: Given, (2z - z) dz along the directed line segment from -i to 1. What is a parametric form of z(t)?…

A: Given integral ∫Γ2z-z¯dz where Γ is the line segment from -i to 1. i.e., it is a line segment from…

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: Which of the following represents a plane surfac whose x-, y-, and z-intercepts are a, b, and c…

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Q: 6. If S is the surface of the sphere x + y² + z? = 9. Prove that curl F- ds = 0. %3D

A: Solution:-

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

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Q: Sketch the following Cartesian Products on the plane R? or in space IR. a. {-1,0,2} x {1,2, 3} b.…

A:

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.