Is velocity a vector quantity? I've read that velocity is a vector quantity. Then why do most of the people when stating the velocity of an object never use direction? Most of them usually say that 'a car is moving at a velocity of 10 m/s'. But that doesn't indicate direction. So, is this usage wrong or is velocity not a vector?
 A: Common usage often does not differentiate between terms which have precise scientific definitions.
Examples are mass & weight, gas & vapour, and the example that you have given, speed & velocity.
This can also spill over into the words used by scientist who will use the term weighing rather than massing although of course many scales compare weights.
In common usage the amount of stuff (mass) is important and that is called the weight and so it is with speed/velocity where distance/time is the quantity being referred to.
Once a term/idea is in common usage it is almost impossible to change it so that it conforms to a precise scientific definition as such a change would not really any effect on the way it is used by the general public except for being inconvenient.
A: *

*Speed is all the police officer cares about when stopping you for speeding. "He's going 100 km/hr! Stop him!"


*But ask NASA whether they care about not just speed but also direction of near-going comets. "It's heading towards Canada with 3000 km/hr!"
Both versions are in use in technical work in different fields. It just depends on the need.
A: When one says, The car is moving with $200\ \text{km/sec}$. That means, they are only concerned with the magnitude of the velocity that is speed.
Whether it's important to specify the direction is a matter of problem. In some cases, the direction plays a role, and in some, it doesn't.
A: Yes, a vector quantity, such as velocity, has a magnitude and (unless the magnitude is zero) a direction.  To fully specify it, you have to indicate both.
But often we only care about one or the other.  Sometimes, e.g. if you want to know whether you're obeying the speed limit, you may only care about the fact that your car is moving at 60 km/h, but not in which direction (at least as long as it's roughly parallel to the road).  At other times, e.g. if you want to know if you turned the right way at the last intersection, you might care about the direction you're going, but not so much about the speed (at least as long as it's not very close to zero).
Now, you may have been taught in school that the magnitude of the velocity vector is called speed, and that it's incorrect to say that "the velocity is 60 km/h" without specifying the direction.  And similarly you may have also been taught that distance is the magnitude of displacement, and that it's incorrect to use the word "distance" for a vector or "displacement" for a scalar quantity.  (For some reason, nobody ever says such things about acceleration or force or momentum, which have no separate names for their magnitudes.)
Honestly, in my opinion those classroom rules are all just so much made up nonsense.  In practice, there's rarely if ever any actual ambiguity in using the same word for a vector and its magnitude, since the magnitude is just a part of the vector (and there's no way to confuse it with the other part of the vector, i.e. the direction).  It's perfectly meaningful to say that your car is a Toyota without also specifying the exact model and year.  And it's just as meaningful to say that your car's velocity is 60 km/h without specifying the direction.
Of course, if you prefer to stick to the rules you learned in school, you can just say that your car's speed is 60 km/h, and be understood just as well.  It's even shorter.  But in practice, people frequently treat the words "speed" and "velocity" as synonyms, and there's nothing wrong with that, at least unless you're a high school physics teacher.

The one context where a potential ambiguity does arise is, in fact, precisely in the context of high school physics — specifically, at the point in the curriculum where true vectors have not yet been introduced as a concept, but signed magnitudes are being used as substitutes for 1-dimensional vectors.
In that specific context it's important to differentiate between a speed (i.e. an unsigned magnitude) of 60 km/h in an unspecified direction and a velocity (i.e. a signed magnitude) of ±60 km/h along either the positive or the negative direction, depending on the sign.  If you don't, you're almost inevitably going to end up with sign errors leading to absurd results, like the relative velocity of two cars colliding head on at 60 km/h each supposedly being 60 km/h − 60 km/h = 0 km/h, rather than the correct (+60 km/h) − (−60 km/h) = +120 km/h.  (Or is it −120 km/h?  Depends on which car you're looking at and which way your positive axis points…)
One could argue that this is even important enough to introduce specific (and somewhat artificial) vocabulary distinctions like "speed" / "velocity" and "distance" / "displacement" for it, and indeed that's presumably why those distinctions were invented and promulgated in the first place.  (And I guess the reason why similar word pairs don't really exist for acceleration and force and momentum is that, in a traditional high school physics curriculum, those concepts are only introduced after moving on from signed magnitudes to actual vectors.)
I'm once again going to respectfully disagree, though.  Pedagogically, spending lots of effort on teaching an artificial terminology distinction that will soon become unnecessary is just wasteful.  In the long run, the really important lesson to teach is the difference between vectors and their magnitudes as concepts, not just that they (sometimes) have different names.  And, of course, also the importance of checking that the results of your calculations, whether inside or outside the classroom, make physical sense.
A: It is a vector quantity, and we almost always indicate direction unless we state Magnitude of Velocity specifically. Direction is usually stated with the help of positive/negative sign (+/-). Positive sign indicates forward direction and negative sign indicates backward direction. Sometimes direction is written with the help of a reference point. (An object is heading towards this place).
In non-technical usage, velocity is often used as a synonym for speed too.
A: Vector quantities do need both direction as well as magnitude to be expressed completely. When we say  'a car is moving at a velocity of 10 m/s' we mean to say that the car(say car A) we are talking about is moving a specific direction ( which is unknown to us). 
Now you might wonder why have we used the term velocity and not the term speed. In most cases in addition to car A there is another car ( say car B) moving in the opposite direction of A with the same speed.[It may come in some question or something idk] Let us add another car c going  with half the velocity of A.
Now let us see the how terms of speed and velocity give us different data :  

If we treat all the values as speed we get:  
Speed of A : 10m/s 
Speed of B : 10m/s 
Speed of C : 5 m/s


If we treat all values as velocity we get
Velocity of A : 10m/s 
Velocity of B : -10m/s 
Velocity of C : 5m/s 
Now if we are only given the information as Velocity of A is 10m/s, Velocity of B is -10m/s and Velocity of C is 5m/s you can now easily assign direction to these cars as A and C go in same direction and B goes in the opposite direction dosent matter if A and C are headed north south east or west.
And if you are talking about people saying velocity of some car is 10m/s then to them velocity is just a cool alternative to speed.
