draw one vector as an arrow with its tail at the start; draw the second vector as an arrow with its tail at the first one's tip (but keep the arrow pointed along the actual vector's direction). If you have several vectors to add, just keep appending them like that, nexttail-to-lasttip. The total vector extends from first tail to the last tip.
Oh you've edited the question so my reply stops making sense :(
To the new questions: what lightgrav says ... add the vectors "head to tail".
I was guessing you knew this - and was working on common mistakes people make.