Find 3rd point in 3D space based on position of 2 points

Question

Assuming i have 2 points $P_1$ and $P_2$ having coordinates of

$P_1 = (x_1, y_1, z_1)$

$P_2 = (x_2, y_2, z_2)$

I want to find the coordinates of a 3rd point ($P_3$) where it creates a straight line if connected with $P_1$ and $P_2$

P3 should be after $P_2$ on the virtually-created-line (aka --$P_1$--$P_2$--$P_3$--) and should have distance $D$ from $P_2$

Thanks!

Answer 1

The solution is $$ \vec{P_2}+D\frac{\vec{P_1P_2}}{|\vec{P_1P_2}|} $$ You take the vector $P_2$, and you add $D$ times the unit (length one) vector that points from $P_1$ towards $P_2$.

Answer 2

1. find a vector that points toward $P_2$ from $P_1$. That's $$\vec v = P_2 - P_1 = (x_2-x_1, y_2-y_1, z_2-z_1)$$
2. find a vector of length 1 in that same direction. This can be done by finding the length $$|\vec v|=\sqrt{x_v^2+y_v^2+z_v^2}$$ and dividing the components of $v$ by it: $$\vec v^* = \frac{\vec v}{|\vec v|}=\left(\frac{x_v}{|\vec v|},\frac{y_v}{|\vec v|},\frac{z_v}{|\vec v|}\right)$$
3. find a vector of length $D$ in that same direction. $$d=D\vec v^*=(Dx_v^*,Dy_v^*,Dx_v^*)$$
4. $P_3$ is now $d$ away from $P_2$. $$P_3=P_2+d=(x_2+x_d,y_2+y_d,z_2+z_d)$$