I have managed to break my X server, and am trying to attempt the solution offered in this page. I only have one user on the system, but I have no idea what its group name is, which means I can't execute the solution offered on the page.

Pressing Ctrl+Alt+F(x) just brings up a flashing white cursor, but I can't type anything. I only have Ctrl+Alt+F7 at my disposal, but unable to login and bring up a prompt with Ctrl+Alt+T.

The only access I have to a shell of any description seems to be by booting into recovery mode and dropping to the shell. However, I then don't get any response from the shell when I type users (as suggested by this page as a way to find out my groupname).

A valid answer to this question would either:

a) Allow me to find out the groupname so that I can execute the sudo chown username:groupname .Xauthority command, or

b) Provide an alternative way to fix the XServer issue, which sounds very similar to the one here.


Question: How to find the distance using dot product from point $A$ to the line through $B$ and $C$?
Answer: The line throguh points $\;B,C\;$ can be put (parametrized) as

$$l_{BC}: B+t\vec{BC}=(3,4,-2)+t(4,-5,7)=(3+4t\,,\,4-5t\,,\,-2+7t)\;,\;\;t\in\Bbb R$$

Thus, a vector from $\;A\;$ to a general point $\;Q\$ on the line is parametrized as

$$\vec{AQ}=(1+4t\,,\,-2-5t\,,\,-3+7t)$$

Finally, we need that 

$$\;0=\langle A\,,\,\vec{AQ}\rangle=2+8t+12+30t-3+7t\implies45t=-11\implies t=-\frac{45}{11}$$

Complete now the exercise.