1. The problem statement, all variables and given/known data In R^2, vectors x = (x1, x2) and a = (a1, a2). For fixed a, det(a, x) is a scalar-valued linear function of the vector x. Thus it can be written as the dot product of x with some fixed vector w. Explain why w is perpendicular to a. Do not use an expression of w in terms of the components of a. 2. Relevant equations Anything involving the dot product, cross product, or determinants I suppose. 3. The attempt at a solution If you simply calculate the determinant, it's clear to see that w = (-a2, a1), and that this is perpendicular to a. I know that in R^3 that det(a, b, x) = (a x b) o x, and (a x b) is perpendicular to a and b, so in the original problem, saying that w is perpendicular to a may be some analog for that. I tried to gain some insight about why (a x b) is perpendicular to a and b follows from the properties of the determinant, but couldn't find anything useful. I also noticed that if we assume that w is perpendicular to x then: w o x = det(a, b) = 0, so x must be parallel to a, which means that w is perpendicular to a. I tried to generalize this, but couldn't come up with anything. I'm stumped. Thanks for your help, and sorry I have no idea how to use LaTex.