How to determine the direction of medium's displacement vectors of a standing wave? Consider the following problem taken from a problem booklet. My questions are:


*

*What is displacement vector?

*And how to determine the direction of displacement vector at a certain point?

*Where is the position with zero displacement vector?



 A: Any material between two nodes is displaced by the same direction. So the direction of B and C has to be the same as well as the direction of A and D due to symmetry. In addition, the direction of A must be the opposite of B since they are across from a node. Similarly the direction of C and D must be opposite.
So the two possible configurations are
   A-->  <--B     <--C        D-->   (figure d)
<--A        B-->     C-->  <--D      (figure c)

The correct answer is (2).
A: Just because a wave is a standing wave doesn't necessarily mean that the particles themselves do not move, in face if the particles themselves didn't move there wouldn't be any wave motion at all. For a longitudinal wave (made of particles that oscillate in the direction of wave propagation like here as sound waves) particles oscillate left and right but have no net displacement.
Take a look at this site on standing waves in compression/longitudinal motion, perhaps it will help you understand what the answer is and why it is the correct one.
http://www.acs.psu.edu/drussell/Demos/StandingWaves/StandingWaves.html
A: A standing wave is a wave that has nodes. The points of the wave go up and down in some places, and remain at zero at others (the nodes). The general form of a standing wave is a sine curve that remains at a fixed position, but its amplitude changes in time between $+A_0$ and $-A_0$. Specifially, there is a time where the wave form is completely flat.
                                                       
                                                                                   (From Wikipedia)
The formula is something like
$$f(x) = A_0\cos(\omega t)\,sin(kx)$$
(not the most general form). Compare to a moving wave which has a fixed amplitude, but a changing offset, so it seems to move along the axis.
$$ f(x) = A_0\sin(\omega t + kx)$$
Now in your case you have a tube with air. Your waves don't go up and down (transversal), but back and forth (longitudinal). The nodes are points where the air doesn't move, anti-nodes are where the air moves maximally. Still, it can be described by the same equation. You can try to draw a sine-curve through your first figure. The $y$ value should be the air displacement at point $x$, at a fixed time ($t=0$ or $t=\pi/\omega$). The sine curve must cross the $x$ axis at the nodes, and have maxima and minima at the antinodes. There are two ways to draw the curve, which are mirrored along the $x$ axis. A positive displacement means that the air molecules are moved to the right (compared to where they should be at $t=(\pi/2)/\omega$), a negative displacement means they are moved to the left. You should be able to read off the correct displacement vectors from your drawing.
A little caveat: Don't confuse displacement and pressure, or speed. The nodes always have zero displacement, but the pressure there changes all the time. The points A, B, C, D (on the slopes of the curve) sometimes have zero displacement, when the waveform crosses the $x$-axis, but at that moment the air has the highest speed (change of displacement).
