WRITTEN HOMEWORK, CALCULUS III, SUMMER SESSION II, 2017(1) Find the equation of the sphere that has center (2,-3,1), and passes through the origin(0,0,0).(2) LetSbe the sphere given by the equationx2+y2+z2-16x+ 8y-8z= 2.(a) Find the center and radius ofS.(b) LetCbe the circle given by the intersection ofSwith theyz-plane. Find the center(which is a point inR3) and radius ofC.(3) There are many spheres that pass through the points (1,1,1) and (3,2,3).Find thesmallest one (in the sense that it has the smallest radius).(4) Find a unit vector in the direction ofh6,3,2i.(5) Let-→v=-2-→i+-→j+ 2-→kand-→w= 3-→i+ 2-→j--→k.(a) Compute the vector projection of-→wonto-→v, proj-→v-→w.(b) Compute the scalar projection of-→wonto-→v, comp-→v-→w.(6) Find the area of the parallelogram with vertices (1,0,2), (2,1,2), (1,3,5), and (2,4,5).(7) For each of the following pairs of lines, determine whether they are parallel (or areidentical), intersect, or are skew.(a)L1satisfies the symmetric equationsx4=y+ 2-2=z-1-4andL2satisfies the symmetric equationsx+ 4-1=y2=z-51.(b)L1can be parametrized by-→r1(t) =h6,2,3i+th3,0,1iandL2can be parameterizedby-→r2(s) =h2,3,-1i+sh-1,1,2i.(8) Find an equation for the plane containing the three pointsP= (1,2,3),Q= (3,1,2),andR= (2,1,3).(9) Find the distance from the plane given by 6x-2y+ 3z-1 = 0 to the point (4,1,3).(10) Reduce the equationx2+y2-z2+2x-4y+4 = 0 to one of the standard forms, classifythe surface, and sketch it.(11) At what point do the curves-→r1(t) =ht,2-t, t2iand-→r2(s) =h1-s, s+1, s2+1iintersect?Find the cosine of the angle of intersection.(12) Find a vector function that parameterizes the intersection of the paraboloidz=x2+y2and the planez= 2y.(13) Find the length of the curve determined by-→r(t) = 2t-→i+√3t2-→j+t3-→k1
fort∈[0,2].(14) Consider the (vector-valued) function-→r(t) =hcost,sint,1-ti,and letCbe the space curve determined by this function.(a) Find parametric equations for the tangent line toCat the point (1,0,1).(b) Find an equation for the osculating plane ofCat the point (1,0,1).(15) Find the curvature (as a function of time) of the curve determined by-→r(t) =h4 cost,9 sint, ti.(16) Suppose that, at timet= 0, a particle with mass 3 has position vector-→r(0) =-3-→j+-→kand velocity-→v(0) = 5-→j-2-→k.The particle is then subjected to a constant force of-→F= 6-→i+ 3-→k.(a) Find the position of the particle (as a function of time).(b) When is the particle moving most slowly?(17) Consider the vector function-→r(t) =ht2-t,2t3, t2i.ComputeaTandaNas functions of time.(18) Consider a particle with position function-→r(t) =ht, t2-3t,2t2i.(a) Find the velocity, acceleration and speed of the particle.