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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?

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    $\begingroup$ You pick the coordinate system, so when you are adding or subtracting vectors the main thing is to preserve how they act, are they in opposite direction, the same direction, or at some angle to each other that you have to split them into components. As far as g is concerned, TBH, I would follow the convention and setup of the guy who will be marking the exam papers : ) . G is positive in the sense that things go faster when they drop, maybe that's the best way of thinking about it. $\endgroup$
    – user108787
    Commented Nov 30, 2016 at 2:39
  • $\begingroup$ Vectors aren't positive or negative. Components of vectors can be positive or negative, but the signs depend on what coordinate system you're using. $\endgroup$
    – user4552
    Commented Jan 10, 2018 at 19:58

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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.

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$g$ 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.

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  • $\begingroup$ It always helps to draw a picture showing the directions of forces, then correlate the signs used in equations (e.g., Newton's 2nd Law equations for free-body diagrams) with those pictured force vectors. Pictures are invaluable in kinematic and dynamic studies. $\endgroup$
    – Bill N
    Commented Jan 10, 2018 at 15:32
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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.

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  • $\begingroup$ No, the number $g$ should always be positive. The sign used when applying Newton's 2nd Law or some of the kinematic equations will be positive or negative depending on the choice of coordinates. For example, for objects in free fall, if up is chosen to be the positive direction, $a_{\mathrm{vert}}=-g$. The negative sign should be explicit and $g$ itself is positive. $\endgroup$
    – Bill N
    Commented Jan 10, 2018 at 15:25
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    $\begingroup$ If you mean to treat $g$ as a vector, then use vector notation, e.g., $\vec{g}$ so that $\vec{g}= -g\hat{j}$ or $g\hat{j}$ depending on the coordinate system. $\endgroup$
    – Bill N
    Commented Jan 10, 2018 at 15:27

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