1. Given the vectors u = i+j+2k = (1,1,2), v 2 2i j+k = {2. l, 1}, and w = i+3k = (1,0, 3) nd: (a (3 points) The dot product 11 - v ( (3 points) The cross product v x w (c) (3 points) The projection of 11 onto v, projvu ( (3 points] The volume of the parallelepiped formed on u, v, and w. 2. (a) (8 points] Are the two lines [1 : :r = 2+t, y = 2+3t, z = 3+t and 2 : :1: = 1+23, y = 2 + s, 2 = 1 +5 parallel1 do they intersect, or are they skew? If they intersect give their point of intersection. (Explain your answer.) (b) (8 points) Find the equation of the plane that contains the line I : :1? = 2 + Ly = 1 11,2 = 3 + t and the point Pg(l.{], l). 3. A curve has parametric equation 1' = t i + th + gt\"? k = (if, 2t, @329). (a) (6 points) Find the unit tangent vector to the curve at the point where t = 1. (b) (4 points] Find the equation of the tangent line to the curve at the point where t = 1. (c) (5 points] Find 3 where s is arc length along the curve. Find the total arc length .9 along the curve from the point where t = U to the point where t = 4. 4. (14 points) Solve the initial value problem to nd 1' = r{t) for all t 2 0, when dr 3 1 _ 1 =t 1 '2' ," k dt 2( J\") \"0 J+t+1 with initial condition r({]) = k. 5. (15 points) Find all rst and second order partial derivatives of f(;r, y) 2 fly? + cos(:ry) + ysinm 6. (15 points) If 'w = my + lnz with v2 :r = , y =1: +1}, 2 = cosu, u use the chain rule to nd the partial derivatives % and 31% at the point where (u 1:) = (1, 2). 7. (12 points) Given the function nme=M+n Find the function's domain and range. Describe or sketch the function's level curves. Find the boundary of the function's domain. Is the domain open, closed, or neither [explain]? Is the domain bounded or unbounded (explain)