Determine equations of nomral line and the tangent plane to parametric surface defined by vector-valued function R (u, v) =< v² – u, u? (0, 2, 1). U, 2u + v > at the point
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- Find a vector function that parameterizes the curve of intersection between the surfaces y^2-z^2=x-2 and y^2+z=9. Use t as the parameter.Consider the surface whose equation is f(x, y) = x^2 - y^2 Determine the vector function r(t) that describes the curve of intersection between f(x,y) and the cylinder of radius 3 that runs along the z axisDetermine the vector equation and the Cartesian equation of the plane tangent to the surface: z = x3 − y3 in the point (0, 1, −1).
- Determine the vector equation and the Cartesian equation of the plane tangent to the surface:z = x3 − y3in the point (0, 1, −1).Identify the type of parametric surface produced by the parametric vector functionr(u,v)=⟨2sinucosv,2sinusinv,2cosu⟩r(u,v)=⟨2sinucosv,2sinusinv,2cosu⟩ with 0≤u≤π2,0≤v≤π20≤u≤π2,0≤v≤π2.Sketch the space curve represented by the intersection of the surfaces x2 + y2 + z2 = 16, xy = 4. Then represent the curve by a vectorvalued function using the parameter x = t (first octant) .
- Find a vector normal to the surface 3z 3 + x 2 y - y 2 x = 1 at P =(1,-1,1) .Let S be the surface obtained by revolving the plane curve 2x=sqrt(9−y2) about the y-axis.Obtain a vector equation for S and use it to find the equation of the tangent plane to S at the point (-1,1,-1).Represent the line segment from P to Q by a vector-valued function and by a set of parametric equations P(0, 0, 0), Q(5, 2, 2).
- Find the equations of the normal line and the tangent plane to the parametric surface defined by the vector-valued functionR(u, v) =< v^2−u, u^2−v, 2u+v > at the point (0,2,1).Compute the equation of the plane tangent to the surface x^2+3 y^2+z^3 = 8 at the point (2,1,1).Two plane curves are defined by the vector functions r1(t)=⟨t,2t−4⟩ and r2(u)=⟨2u,u2⟩. Find the angle α between the curves at the point of their intersection.