Help Please will all parts on paper

1) Given the below four points in space, verify that they are the vertices of a parallelogram E 50 pts each and find the area Choose one of the below. (0, 0, 0) (3, 2,4) (5, 1,4) (2, -1,0) 1) Find the volume of the region below the plane 3x + y + z = 3 and above the xy plane in 2) Given the below four points in space, verify that they are coplanar or find the area of the the first octant. You may use either a double integral or a triple integral. parallelpiped formed by the vectors between the points. (0, 1,2) (2, 0, 1) (1, -1,1) (-1, -1,0) 2) Use a triple integral to find the volume of the region under the plane 6x + By + z = 12 in the first octant. 3) Prove that the cross product of any vector with itself yields 0. You may not simply choose random numbers for vector components - you must perform the proof in symbol 3) Convert to cylindrical coordinates and evaluate. form. CL x dady dx 4) Convert to spherical coordinates and evaluate. If forty +7 dady da 1) Let r(t) = (2t, 6 - t?). Find v(t) and a (t) at the point (2,5). C 2) Find unit tangent vector T(t) at the point (6V2, 6V/2,3) to r(t) = (12 cost, 12 sin t,3). Let u = (4,14,-2) v = (-5,2,4) w = (2,0, -5) be vectors in space. 3) Find principal unit normal vector N(t) att = = to r(t) = (5 cost,5 sint). 1) Find the angle between the x-axis and u 10 pis 4) Find normal component of acceleration an(t) at t = = to r(t) = (4 cost, 4 sin t, 5t). 5) Find tangential component of acceleration ar(t) att = 2 to r(t) = (6t - 7,t2, -10). 2) Find the angle between the y-axis and u 10 pts Round your answer to four decimal places. 6) Find the length of the curve given by r(t) = (2t, 5 cost ,5 sint) on [0,3]. 7) Find curvature Kof r(t) = (t, 2t2, 2t). 3) Find the angle between the z-axis and v 10 pts 4) Find proj.w 10 pts