As far as my knowledge is concerned, a vector quantity should possess magnitude and direction & more over it should also obey the laws of vector addition.
As we all know that the vector sum of 3 newtons in the x direction and 4 newtons in the y direction will acting at a point will produce a resultant of 5 newtons. Where did the remaining 2 newtons go?
I mean we have applied a total of 7 newtons of force on a point sized particle but the output is only 5 newtons, so it appears as if a 2 newton force is disappearing here. In which other form does it reappear rather than the resultant? Or is it something like the remaining 2 newtons of force just vanishes and it doesn't appears in any other form?
Please correct me if I am going wrong on this issue.
Best Answer
There are some quantities that only come with one sign, for example mass/energy. So in any process if you start with some mass/energy $m$ then whatever happens you can't end up with less than $m$.
But other quantities come in both positive and negative magnitudes, and these will cancel. If you add a velocity of 10 m/s North to a velocity of 10 m/sec South then obviously they will cancel and sum is zero. You don't ask "hey, what happened to the 20 m/s I started with" because we all know equal and opposite velocities cancel. Likewise accelerations. If you add an acceleration of 9.81 m/s$^2$ up to an acceleration of 9.81 m/s$^2$ down then the total acceleration is zero. Again you wouldn't be puzzled as to where the 19.62 m/s$^2$ you started with have gone.
But remember that force is just mass times acceleration (it's one of Newton's laws though I always forget which). So if accelerations cancel it shouldn't be a surprise that forces cancel as well.
To see why you're getting cancellation in your example consider the simpler example of adding two 1N forces at right angles to produce a force of $\sqrt{2}$ N:
To see what is going on first rotate the diagram 45ยบ:
and then split the vectors $F_a$ and $F_b$ that you're adding together into $x$ and $y$ components:
Now it should be obvious what is going on. The total $y$ component of $F_a + F_b$ is $F_{ay} + F_{by}$, and these components point in the same direction so they add together. The total $x$ component of $F_a + F_b$ is $F_{ax} + F_{bx}$, but these components point in opposite directions so they cancel and sum to zero.
And that's how two 1N forces can sum to less than 2N. It's because the two forces have a components with the same sign that sum and components with opposite signs that cancel.