Let A = (1, 0, 4) and B = (7, 0, 4). Find an equation for the sphere consisting of all points P = (x, y, z) such that the vector AP is orthogonal to the vector BP. What are its center and radius?

Question 2 Find a b if (a). a =< 1, 2, 1 >, b =< 0, 3, 0 >. (b). a =< 1, 2, 4 >, b =< 2, 4, 8 >.

Question 3 (a). Find a vector perpendicular to the plane containing the points P = (1, 2, 1), Q = (4, 2, 0), and R = (3, 2, 3). (b). Find the area of the triangle with vertices P = (1, 2, 1), Q = (4, 2, 0), and R = (3, 2, 3). Question 4 (a). Find a vector v such that < 1, 0, 1 > v =< 2, 3, 2 >, or explain why there is no such vector. (b). Find a vector v such that < 1, 2, 3 > v =< 2, 3, 2 >, or explain why there is no such vector.

Question 5 Consider the points A = (2, 1, 5), B = (4, 2, 4), C = (3, 2, 7), and D = (1, 0, 4). Find the volume of the parallelepiped with adjacent edges AB, AC, and AD.