I need help in answering these questions. can you please make the answer a bit brief but detailed and if required drawing please feel free to do so for more illustration.

thank you!

1. You have a data set consisting of points that lie along a plane. How do you find the normal vector to the plane? Provide a real-world example where finding the normal vector would be useful, and explain why.
2. Consider the sphere of radius 3, centered at the origin. Parametrize its intersection with the first octant, in two different ways. STUDY TIP: COORDINATES, AND GRAPH
3. Suppose you know the location of a curve at time t=0, as well as its acceleration vector at all times between t=0 and t=1. How do you compute its location at time t=1? Provide a real-world example where this is useful.
4. What are the names for the first, second, and third derivative of position? Provide an example where the third derivative of position is useful.
5. How do you compute the length of a curve? Suppose you are also provided with a cost function at every point in space. How do you compute the total cost of a curve?
6. You have a parallelepiped and know the coordinates of one corner and vectors providing the three adjacent corners. How do you compute the volume of the parallelepiped?