Find the unit tangent vector T(t) and find a set of parametric equations for the line tangent to the space curve at point P. r(t) = (3 cost, 3 sin t, 3), P(\\{. VZ: 3) Part 1 of 5 The unit tangent vector T(t) at t is defined as follows: T(t) = - r't | r'(t). r'(t) # 0 First find the derivative of r(t). r'(t) = (-3 sin (t) ,3 cos(t),0 (-3 sin(t), 3 cos(t), 0) Part 2 of 5 Now find the magnitude of r'(t). [|~'(t) |) = (-3 sin t)2 + (3 cos t)2 + (0)2 = V 9 sin2 t + 9 cost = 3 4 P 3 Part 3 of 5 The unit tangent vector T(t) at t is as follows: T(t) = (-3 sin t, 3 cos t, 0) 3 (-3 sin (t),3 cos(t),0) (-3 sin(t), 3 cos(t), 0) Part 4 of 5 Verify that t = # at the point P( 7, 7/7, 3). That is, evaluate r(#). Find the unit tangent vector at t = It. T (t) = _ -3 sin t, 3 cos t, 0) T(I) = 3-3 sin(), 3 cos ( ). . 3 3 V 2' V 2 ,0 1 1 VIV20 Part 5 of 5 The set of parametric equations for the line tangent to the curve r(t) at point P(x0, Yor 20) is given by the following. x = X1 + at y = y1 + bt a, b, and c are direction numbers z = 21 + ct Use the direction numbers a = b = 3 VZ and c = 0 to find the set of parametric equations for the line 3 tangent to the curve r(t) at point P VZ, 3). (Enter your answers as a comma-separated list.)Find v(t), a(t), T(t), and N(t) (if it exists) for an object moving along the path given by the vector-valued function r(t). (If an answer is undefined, enter UNDEFINED.) r(t) = 3ti - 3tj V(t) = a(t) = T(t) = N(t) = Use the results to determine the form of the path. O quadratic O hyperbolic O sinusoidal O straight line Is the speed of the object constant or changing? O constant O changing O can not be determined