3d vector postions

Question

Cartesian coordinates...

Given a starting position of $x_1=0, y_1=0, z_1=0$ travelling to a destination position of $x_2=500,y_2=500, z_2=500$.

I can calculate the length of the connecting vector(I know how to do this)

I want to calculate the coordinates of the position $x_3, y_3, z_3$ after travelling distance (d) down the vector from position 1 to position 2

I have standard o-level maths, I can apply a formula in programming, but this level of geometry is beyond my grasp.

Answer 1

The traveling vector is given by the parametric equation

$$OP=(x_0,y_0,z_0)+t(x_2,y_2,z_2)=t(500,500,500)$$

with $t\in[0,1]$.

Answer 2

The answer to your first question:

The length of the connecting vector is given by $$l = \Vert (x_1, y_1, z_1) - (x_2, y_2, z_2) \Vert_{2} = \sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2} + (z_{1} - z_{2})^{2}} = \sqrt{3 \times 500^{2}} = 866.025403784$$