# Calculate direction vector given start-end positions and velocity?

I'm trying for a week now and I really have a hard time with this one. I'm pretty sure that this is something very simple but I can't seem to find a solution. I'm making a game where the enemy is shooting arrows to the player without including air friction.

We already know the following values (just some random values):

1. Enemy position [startPoint = (0,0)].
2. Player position [endPoint = (20,10)].
3. Gravity [g = 10] (In-game physics are using this value).
4. Velocity [Vo = 25].

The question is "How to calculate the direction vector [u] using these values?".

I have already used u = 2Δ - gt² / 2Vo*t where Δ = endPoint - startPoint but it doesn't give me the correct vector which later will be multiplied with velocity.
I can see that because, by finding the angle of the vector θ = atan2(u.y, u.x) and getting back the direction vector u = (cosθ, sinθ), the vector is not the same.

After u is known, I can find the starting horizontal [Vox] and vertical [Voy] velocities and apply them to the arrow by doing:
Vox = Vo * u.x
Voy = Vo * u.y

• You know the required x-distance to travel, the total speed ( $\mid Vo\mid$ ) , and the relationship between initial vertical speed and time before impact w/ the ground. Solve that set of equations. Sep 5, 2014 at 13:27
• I have already done that with a target that is on the ground. The problem starts when I want to hit a target that is not on the ground, like the endpoint we have here which is y = 10 above the startPoint. I'm using positive y from the startPoint and up. Sep 5, 2014 at 13:54
• It's the same equations, except you solve for the time to fall back to "start height + 10 meters" Sep 5, 2014 at 16:52
• Sorry but I can't understand what you're saying. I'm not good in physics at all and I'm learning for a week now. If it's possible I would like to know the steps involved for the solution of this problem. Sep 6, 2014 at 13:27

Finally after more than a week I managed to solve the problem. Here's the solution that a lot of people are seeking (I'm sorry for the expression of the formulas):

First we find the delta vector:
delta = endPoint - startPoint = (20, 10)

There are two angles involved, the inclination angle [θinc] and the projectile angle [θproj].

The inclination angle is:
θinc = atan2(-delta.y, delta.x) = 26.56

The projectile angle is:
θproj = atan(Vo² + √(Vo^4 - g(g·delta.x² + 2·delta.y·Vo²)) / g·delta.x) = 79.69
θproj = 90 - θproj = 10.31

We add the two angles and we have our final angle:
θ = θinc + θproj = 36,87

Now we can find the direction vector:
u = (cosθ, sinθ) = (0.8, 0.6)

And we have our horizontal and vertical velocities:
Vox = Vo * u.x = 20
Voy = Vo * u.y = 15

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