Let $L \subset \mathbb{R}^2$ be a line through the origin, and let b $\in \mathbb{R}^2$ be any point.
a.) Find a geometrical construction of the closest point v $\in L$ to b when the distance is measured in the standard Euclidean norm.
b.) Use your construction to prove that there is one and only one closest point.
c.) Show that if $0 \ne$ a $\in L$, the the distance equals $\frac{\sqrt{||a||^2||b||^2 - (a * b)^2}}{\|a||} = \frac{|a\times b|}{||a||}$.
My attempt:
a.)Let $l_1,l_2$ be a basis for L. Then the general element of $v \in L$ is a linear combination of the basis vectors. Thus, $x_1l_1 + x_2l_2 = Ax$ is the m x n matrix formed by the basis vectors and x = $(x_1,x_2)^T$ are the coordinates of v. So, the closest point in L to b is $||v-b||^2 = ||Ax - b||^2$ over all possible $x \in \mathbb{R}^n$.
b.) I do not know how to do this
c.) I do not know how to do this