Detailed Explanation for all parts please in neat writing on paper i have there final answers just need to now there methods Step by Step Solution.

1) Given the below four points in space, verify that they are the vertices of a parallelogram 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 + 8y + z = 12 in the first octant. 3) Prove that the cross product of any vector with itself yields 0. You may not simply 3) Convert to cylindrical coordinates and evaluate. choose random numbers for vector components - you must perform the proof in symbol form. I'll x dz dy dx 4) Convert to spherical coordinates and evaluate. Matty' +7' dz dy dx 1) Let r(t) = (2t, 6 - t2). Find v(t) and a (t) at the point (2,5). 2) Find unit tangent vector T(t) at the point (6V2, 6V2,3) 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 pts 4) Find normal component of acceleration an(t) at t = = to r(t) = (4 cost, 4 sint, 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 projuw 10 pts