What's the physical meaning of change in momentum vector? If I there is a initial momentum of 10Ns upwards, and final momentum of 10Ns to the right, I can find the difference in momentum by drawing a triangle and finding the resultant vector. But, how is there a change in momentum if both are 10Ns? I know the direction changed but the change in momentum using pythagorean theorem would be $\sqrt{200}$Ns, what physical meaning does that even have?
 A: Since momentum is a vector, the quantity being measured did indeed change. As you state, while the magnitude is a constant $10\ \mathtt{kg\cdot m/s}$, the direction has altered. But what does that mean?
The meaning of this change lies in understanding the distinction between momentum and impulse. The change of momentum is called an impulse. The common English definition of impulse is a "drive or acting force, an impetus." In physics, the meaning is similar:
$$\Delta \mathbf{p} = \mathbf{F}\Delta t$$
Rearranging the above statement and limiting our time period gives us the differential form seen below. Now we note that the change in a system's momentum during a time period is equal to the force exerted on the system!
$$\mathbf{F} = \dfrac{d\mathbf{p}}{dt}$$
Now the above statement should smack of $\mathbf{F} = m\mathbf{a}$. It's really not so different. We're merely restating Newtown's 2nd Law. The force on a system is proportional to its mass and acceleration. The force on a system is proportional to a change in its momentum: which is its mass and its changing velocity, acceleration.
