The position of a particle in the xy-plane at time t is r(t) = (t+ 1)i + (1" - 2) j. Find an equation in x and y whose graph is the path of the particle. Then find the particle's velocity and acceleration vectors at t = 3. The equation for the path of the particle is y = (1) The equation r(t) = (6 cost) 1+ (3 sin ?) | + (2t) k is the position of a particle in space at time t. Find the particle's velocity and acceleration vectors. Then write the particle's velocity at t =0 as a product of its speed and direction. The velocity vector is v(1) = ( ) . . ( D ) + ( D * (2) The equation r(t) = (2t+ 3) + (v5t)j+ (21") k is the position of a particle in space at time t. Find the angle between the velocity and acceleration vectors at time t = 0. The angle is radians. (3) (Type an exact answer, using x as needed.) The tangent line to a smooth curve r(t) = f(t)1 + g(t) + h(t)k at t = to is the line that passes through the point (!(to) .9(to) h(') ) parallel to v (to) . the curve's velocity vector at to- User (to) and r (to) to find paramet line that is tang he given curve at the given parameter value t = to. 1+ 7+ 8 1+ tIn 7 K. 6- 7 Find the sta tion for the tangent line. (4) OAXEmy O B. XM OC. XBY OD. X= 7. y" 750. 2# 7 The tangent line to a smooth curve r(t) = 1(t)1 + g(t)| + h(t)k at t=to is the line that passes through the point (!(to) . (to) .h (to) ) parallel to v (to) . the curve's velocity vector at to- Use r (to) and r' (!) to find param line that is tangent to the given curve at the given parameter value t = to. r(t) = (cost)i + (sint) +(sin 80)k. 42 Find the standard parametrization for the tangent line. (5) OA X= - 1, y=t, z= 8t OB. X"Ly= - 1,2= - 8t OC. x= -1,yst.z= - 8t OD. x=ty= - 1,z- 81 Show that the vector-valued function shown below describes the motion of a particle moving in a circle of radius 1 centered at the point (3,2,4) and lying in the plane 2x + 2y - 42= - 6. First write the three parametric equations for x, y, and z. What is the parametric equation for x? (6) x = (Type an exact answer, using radicals as needed.)