I'm a high school student and I'm studying Newton's second Law. While my teacher is calculating a net force of an object he always treat all vectors as positive numbers. I think what he meant is the magnitude of the vector which is always positive, but the problem is that he didn't use the magnitude symbol, for example like $||Fg||$. He said the acceleration due to gravity , g, is positive, but shouldn't it be negative since it is pointing down. This also cause a problem: every time when I calculate the net force I can't add all forces together, instead I need to choose to use minus sign when the vector is pointing down. So in general which way is correct?
The teacher's point is that all vector magnitudes are positive, and you only add signs because of directions.
So yes, you are right. There will be minus signs as well.
But you can't say that for example $g$ should be negative. It is a positive value on itself, and it only becomes negative, if the direction is opposite to the axis. If you in some specific case choose your coordinate system to point downwards, then the $g$ is positive.
If you had been taught that $g$ is always negative, you might have been confused in that case.
$g$ is somewhat like a fundamental constant. It's a number that we all agree on. All of these are magnitudes, and we put signs in front of them when we need to specify a sense or direction. For example, the charge on a proton is $+e$ and that of an electron is $-e$. There is some confusion because we don't normally write the positive sign explicitly, so in our conventional notation we don't have a way to distinguish a magnitude from a positive signed quantity.
if you take downward direction is positive then g is positive;if you take upward as positive then g is negative.g is a vector that always point down,but it's +-sign is just depends on the direction you take positive.I think your teacher always takes down as + so g is +in this case,but you may decide your own way.However,g always have a magnitude of 9.81,number won't change in any cases, just +-sign.