Let r(u, v)= [2usin(v), 2ucos(v), 2] be a parametrization of a surface S, where 0

Q: Consider the surface S given by: S : r(u, v) = (uv, u² + v³, 3v²), u = [0, 2], v = [0, ³/] whose…

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Q: Evaluate F. dS , where F = and S is the helicoid with vector equation r(u, v) = 0 < u < 1, 0 < v<T…

A: since, y,-x,z3 and, ru,v=u cosv, u sinv,v So, F⇀ru,v=u cosv,-u sinv, v3 Now the normal upward vector…

Q: 1 Find the cross product uxv. U=it 3-k, v=2i-j tk 2. Find the area of AABC and the equations of the…

A: 1. Cross Product of u×v  : To find the cross product u×v for the vectors u=i+3j−k and v=2i−j+k, we…

Q: find the normal vector of each plane plane 1: 2x-3y+3z=4 plane 2: 2x-y+z=1

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Q: 9. State a vector and scalar equation for the plane passing through Po= (xo, yo, zo) with normal (a,…

A: Given:A plane passing through the point P0=x0, y0, z0. The normal vector to plane is n→ = a,b,c.…

Q: The surface S is defined by x + y² + z³ = 3. Find a vector normal to S at the point A with position…

A: Given a surface S the normal vector to the surface at the point A is given by

Q: 5. Find all vectors such that x v = . 6. Find parametric equations for the line through (2,4,-2)…

A: (6) Given :- The parametric equations for the line through 2,4,-2 that is perpendicular to plane x…

Q: The tangent plane at a point Po(f (uovo)g (uovo),h (uo.vo)) on a parametrized surface r(u,v) =…

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Q: Let ri = xi+yıj+z¡k, r, = 1,i+ Yzj+ zzk and r3 = 13i+ Yzj+ zzk be the position vectors of points…

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Q: Let L be the line with parametric equations x = 2+t, y = 5-2t, z = 4 + 3t. Let S be the plane that…

A: Given L is the line with parametric equations x = 2 + t , y = 5-2t , z = 4 + 3t . Let S be the plane…

Q: Draw a picture of two spheres of different sizes being tangent to each other at exactly one point P…

A: For part 1 we are given two spheres of different sizes i.e different radius which are tangent to…

Q: 1. Find the unit normal vector to the surface z = Vx2 + y? at the point (3, 4, 5).

A: A detailed solution is given below

Q: Let r(u, v)= [2ucos(v), 2usin(v), 2] be a parametrization of a surface S, where 0su33,0svS2 T. Then…

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Q: that Let P be the point (1,0,2). Consider the plane through the points A= (3,4,-1), B= (1,-3,3) and…

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Q: 3. Determine vector and parametric equations for the plane through the point (1, -2, 3) and parallel…

A: Here we are given a point and perpendicular direction. We will express equation of plane in vector…

Q: 2. Consider the vector-valued function 12t3 – 72t2 + 96t 15t – 60 6t – 24, if t > 0, t + 4, Ř(t = 4…

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Q: Let r(u, v)= [2usin(v), 2ucos(v), 2] be a parametrization of a surface S, where 0sus3,0svs2n. Then…

A: Parametrization of a surface

Q: R(u, v) = (2v + cos u, 2u + sin v, uv)

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Q: 19. Suppose that the normal unit vector N of a surface S at z = f(x, y) is determined, how does the…

A: Note: As company policy, we solve first question only. Please resubmit the rest of the questions in…

Q: True or False: The vector (10, 2, –4) is normal to the surface x2 + y? – 22 = 22 at the point P =…

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Q: 8. A line initiated from (1,0,2) and parallel to a vector i+j+k what is the parametric equation of…

A: we have to find equation of line in 8.

Q: Consider the ray r(t)=o+ td with origin o=(1, 1, 0) and direction d = (1, 1, 1), and the plane N-x=…

A: To find the point of intersection of the ray and the plane. Given: equation of ray is: r(t)=o+td…

Q: This surface (dreamed up by Steiner while in Rome) looks smooth but is non-orientable, with the…

A: Given: The parametrization for the surface is given by σ(u,v)=x,y,z=sin2usinv,sin2ucosv,cos2usin2v…

Q: a. Find vectors nį and n2 that are normals to II and IL respectively and explain how you can tell…

A: The planes are given by π1: x1x2x3=0-33+λ11-12+λ2313 and π2: x1x2x3=000+μ112-1+μ2340. For π1:…

Q: Let r(u, v)= [2usin(v), 2ucos(v), 2] be a parametrization of a surface S, where 0sus2,0svs27. Then…

A: Given surface is  ,

Q: Find the surface normal vector to the surface z 4-(x +y²) at the point (1, 1,v2) by (a) Vf (b) N =…

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Q: 12. Calculate fw, where S is the portion of the paraboloid 2-x² + y² with 0 ≤ ≤4, oriented by…

A: Answer given below.

Q: Consider the plane that contains the point P(-1,0,1) and is normal to the vector n(1,1,1). Find…

A: Please check step 2 and 3 for solution.