Yes, that's right this is no mathematical formula to state where some unknown quantaty is a scalar or vector. It's more or less a common sense issue I believe. This of the tension in a string holding up a light, obviously its a force but it needs a direction. It cannot exert 10 Newtons downwards to hold the light up. We know whatever the magntiude that if this light is hanging vertically, the tension must act upwards, if we can associate a direction to any quantity, then it is a vector. The velocity of an object, its momentum, a displacement, a trajectory, a force, these are all vectors. A scalar is something more like 3 apples or $1. I don't really have to say I have 3 apples arrrange in a triangle or $1 bill folded upside down. I guess you could if you wanted but we really can't associate a directio to a bill that's folded upside down or a triangular arrangement of apples. We do know some things though, any dot product between to vectors is ALWAYS a scalar. The magnitude of any vector is ALWAYS a scalar. The cross product between two vectors is ALWAYS a vector. This is actually why the angular momentum of a particle is a vector. It is defined as the cross product of the position vector a particle relative to some basepoint crossed into its velocity as observed by someone at that common basepoint. This cross product is then just simply multiply by the mass an in the end we have angular momentum. What about kinetic energy 1/2mv^2? Looks like a vector from that defintion right, it contains velocity and velocity is av ector, so why is energy a scalar. We'll 1/2mv^2 is actually a watered down version of what KE really is. KE is actually defined as 1/2m*v DOT v. The velocities are dotted together to form a scalar, a scalar multiplied by another scalar such as the mass, is ALWAYS a scalar, hope this helps
