TITLE: Shortest length that a vector can have
QUESTION [5 upvotes]: I came to the following question from a past exam: The vector $v = (k, k, 3 − k)$ depends on a variable $k$.
What is the shortest length of the vector $v$ can have? I know that the answer is $\sqrt{6}$, but why...? How do I proceed here to find the result..

REPLY [4 votes]: The length is going to be $\sqrt{k^2+k^2+(3-k)^2}= \sqrt{3k^2-6k+9}$ now use calculus to find the minimum of the function $3k^2-6k+9$ find the derivative is $6k-6$ so if $6k-6=0$ then $k=1$ substitute to find the minimum distance is 
$\sqrt 6$ as desired. Regards