Another thing that vectors can do for us is handle acceleration. We already talked about moving at a certain velocity (distance and direction) but what happens if we want to continue on that path or speed up or turn in a new direction. We have to accelerate or slow down or have force applied to us in a new direction. If our object has a position P and a current velocity of [3,3] which means at every timestep we move 3 in the x direction and 3 in the y direction. If we want to speed up to [10,10] we could of course just set our velocity to [10,10], but that is very unrealistic. That would be like getting in your car and going 30 miles per hour, pressing the gas and in one microsecond be going 100 mph! The sheer force of that acceleration alone would probably kill you. Depending on your mode of transportation you have limits to how fast you can accelerate. This is defined by how much mass you have and how much force you can exert. Because if you took high-school physics, you may remember this.

F=ma

Which means: Force = Mass x Acceleration. This is Newton's Second Law of Motion. We would know the mass of our object ( say your vehicle weighs ton in english measurements. Which is approx 907kg) and we decide it should be able to go from 0 to 30 meters per second (which is about 60 miles per hour, use google to do your conversions!) in 10 seconds. Now we will assume for our simulation that acceleration is constant, that means every second we would need to go 3 meters per second faster. So we want to acellerate at a maximum of 3 m/s^2. So our maximum force is

907kg * 3 m/s^2 = 2721 Newtons or (kg*m/s^2)

But how do we apply this to vectors?

Well first we need to know our desired velocity. In my car example before we were going 30 mph and our desired velocity was 100mph (naughty speeder) That is velocity in 1 direction. Now lets think in 2. our current velocity is the vector [0,0] and we want to go [10,10] we cannot just jump from one to the other. In this case we can actually use the length of the vector to represent the force we would have to apply to go from one to the other. In this case the length of the vector is 14.14 well, say we determine that our maximum force is a vector length of 2, we truncate our desired vector to a length of 2 (we leave it alone if it is less than that already) and that gives us a vector which is the maximum force vector which we can apply to our object. So now that we have force, we divide that by the mass of our object say 10kg and that gives us our acceleration vector. Now for each timestep we do this calculation and add our acceleration vector to our current velocity to get our new accelerated velocity. We do have to truncate our velocity to our maximum speed since the length of our velocity vector is equal to our speed. Then we simply add our velocity vector to our position point and we have our new position for that timestep.

So some things we learned about vector lengths...

Length of the Vector between 2 velocity vectors is the total force required to accelerate from one to the other.

Then length of the velocity vector is the current speed

The Acceleration vector is gained by Dividing the Force vector by the mass of the object

The Acceleration vector is added to the velocity vector each time step to gain a new velocity.

Did I get something wrong? If you know more about this than me, please contribute by commenting below!