Any dependence or independence between vectors is more a matter of the physics they represent than of mathematics. The mathematics are chosen to model the physics of the processes at work.
In the case of projectile motion gravity acts on the projectile and the direction of gravitational force is always vertical. If we ignore air resistance forces, then the vertical motion is basically modelled by the application of downward acceleration = g or upward acceleration = d²y/dt² = -g where y is vertical height. The horizontal acceleration is always 0 or horizontal acceleration = d²x/dt² = 0 where x is the horizontal coordinate. If you were to consider air resistance in the physics then the vertical and horizontal vectors would still be linearly independent but not totally independent as above.

Answer #2 | 23/12 2013 08:37

In the mathematical sense, the linear independence of a set of vectors is defined by the fact that no linear combination of them adds up to zero. If you have only two vectors, this boils down to saying that neither vector is a scalar multiple of the other. So long as the horizontal and vertical velocities are both non-zero, it's clear that neither of them can be represented by just multiplying the other one by some number.

Answer #1| 23/12 2013 04:37