Linear Algebra, Least squares I'm going through the Linear Algebra course on Brilliant.org, but can't figure out how I should know that $\frac{17}{3}$ is the best value for $c$ to fill in here. 
My two questions are:
1) Why do we want to minimize the distance of 
$$3c^2-34c+101$$
2) How can you derive where this distance is minimised? 

You wish to minimize the distance of $c\pmatrix{1 \\ 1 \\ 1} -2 \pmatrix{1 \\ 2 \\ 3}$ to the vector $\pmatrix{5 \\ 0 \\ 0}$.
We have
$$\left\|c\pmatrix{1 \\ 1 \\ 1} -2 \pmatrix{1 \\ 2 \\ 3} - \pmatrix{5 \\ 0 \\ 0}\right\|^2 = \left\|\pmatrix{c-7 \\ c-4 \\ c-6}\right\|^2 = (c-7)^2+(c-4)^2+(c-6)^2 = 3c^2-34c+101$$
To minimize the polynomial $3c^2-34c+101$, write it as 
$$3c^2-34c+101 = 3\underbrace{\left(c - \frac{17}3\right)^2}_{\ge 0} + \frac{14}3$$
so the minimum is attained for $c = \frac{17}3$ and it is equal to $\frac{14}3$.
