As with any vector you need a direction in addition to a magnitude (as you've pointed out). A standard way to express this information is by providing components relative to a coordinate system. Forget about the air-plane for the moment. A vector in the mathematical-plane can be represented by the pair (Vx, Vy) where Vj is the projection of the vector onto the j-th axis. The vector is a mathematical abstraction frequently described as an equivalence class of directed line segments. The "directed line segment" is the arrow. In 3-dim you need three components (Vx, Vy, Vz). Any set of coordinates will do just fine and they don't have to be Cartesian, (x, y, x), they can be spherical, cylindrical etc.
Now back to your aircraft example. If you are interested in aerospace dynamics there are several possible choices of coordinates. Someone mentioned "East" as an example. But this carries some ambiguity in terms of setting up Newton's laws and solving for the motion of the object under a force load. The reason is that at every point on the earth there is a local definition of East and they point in different directions in 3-dim space!
One possibility is something called Earth Centered Inertial (ECI). This is a coordinate system with (0,0,0) at the center of the earth, z-axis through the North pole, and x-y plane at the equator. It is "inertial" because it does not rotate with the earth.
A second possibility is Earth Centered Earth Fixed (ECEF). This is similar to the ECI except the coordinate rotate with the earth so that the x-axis remains at a fixed point.
It is not easy to think in terms of those coordinates. We live in local frames in small neighborhoods on the earth and when we measure a plane's velocity it is relative to us so we are moving too in ECI. Any person can use what is called an ENU frame which stands for East-North-Up. It is a Cartesian frame like (x, y, z) but there is a different ENU at every point on the earth.
Lastly, but not at all least, is a curvilinear set of coordinates called Lat-Long-Alt. You specify the latitude, longitude and altitude of the the object. Its velocity can be expressed in terms of the derivative of these quantities. The directions will coincide with ENU but the values will be scaled. Specifying East only ignores possible rise and descent of the airplane.
Why the variety? Well, some coordinates are more natural for collecting and expressing data and others for expressing Newton's laws. If all you want to do is describe what you see then ENU is natural. Everything is relative to you. But the laws of physics are best expressed in an internal frame and when trying to describe multiple objects as seen by multiple people it is better to have all the data in a common frame, ECI or ECEF. If you're solving Newton's equations for the motion you either need to be in an inertial frame or account for the "fictitious forces" due to being in an accelerated (rotating) frame.