In this problem, let D(x, y) denote the determinant of the 2 2 matrix with rows x and y. Assume the vectors v1, v2, v3 R2 are pairwise linearly independent. a. Prove that D(v2, v3)v1 + D(v3, v1)v2 + D(v1, v2)v3 = 0. (Hint: Write v1 as a linear combination of v2 and v3 and use Cramers Rule to solve for the coefficients.) b. Now suppose a1, a2, a3 R2 and for i = 1, 2, 3, let _i be the line in the plane passing through ai with direction vector vi . Prove that the three lines have a point in common if and only if D(a1, v1)D(v2, v3) + D(a2, v2)D(v3, v1) + D(a3, v3)D(v1, v2) = 0. (Hint: Use Cramers Rule to get an equation that says that the point of intersection of _1 and _2 lies on _3.)

Calculus 1 Chapter 2, Section 2 – Intro to Limits (cont.) and Their Properties Prior to this, were learning how to solve limits as x approaches a number analytically with use of algebra, but now we are going to look at how to solve a limit using a table method. Don’t worryit is not anything hard, it concludesofmaking a table andwriting down close numbers to the leftandrightofthe limitandpluggingthosevaluesto findvaluesthatdrawclosetothelimit.Thisapretty simple method to follow. Let’s look at an example which we can solve for the limit using the table method: Ex.1: − 7 + 12 () = lim = →3 − 3 First, we mus