Consider the unit sphere U: x² + y²+z² = 1 What is its volume? What matrix A changes U to the ellipsoid E: a +8² + 2 = 1? Find the volume of this ellipsoid using determinants. a²

1a.  Find the area of ​​a plane R which has the vertices A(0, 0), B(2, 5), C(6, 3), D(4, -5), and E(0, -4)!​ Hint: Use the broad concept of determinant interpretation 1b.  Find the equation of the plane that passes through the points A(2,0,2), B(0,4,4), and C(1,1,0)!​ Hint: Something to do with normal vectors! Use cross products and dot products.

Illustrate the multiplying any row (or column) by a scalar do to the determinant?

Stable molecules form when atoms arrange in a way to exert equal force on each other. One such molecule is phosphate, PO³-, which forms a tetrahedron shape. The notation indicates 4 oxygen atoms with a single phosphorus atom (and an overall excess of 3 electrons). Consider a coordinate system where the four oxygen atoms that form the tetrahedron are placed at (1, 0, 0), (0, 1, 0), (0, 0, 1), and (−1/3, -1/3, -1/3). (a) To have equal forces the phosphate atom must be placed equidistant from each of the oxygen atoms. Assume the phosphorus atom has the same value for all coordinates, i.e. it is located at (a, a, a). Find a. (b) Determine the bond angle, that is the angle between the lines that join the phosphorus atom to two of the oxygen atoms. Estimate it in degrees.