How to translate a 3d point along a line?

Question

I have two points (vectors) in 3d space. Given some value I want to translate point A by that value along the line that connects point A and B. Is there a simple formula for this? I'm trying to implement it in python.

Answer 1

Let $a,b$ be the two vectors in question. We can define a third vector $d=b-a$. This vector will point from $a$ to $b$. Normalize this to the unit vector $n$:

$$n=\frac{d}{|d|}$$

Now, if you have some distance $\lambda$ you want to move towards $b$ you create the new point $a'$ as

$$a' = a + \lambda n$$

Answer 2

Any point $M$ on the line going through $A$ and $B$ can be represented as $$M = A + t\cdot \overrightarrow{AB}$$ where $t$ is a real number. For instance, $t=0$ gets you $M=A$, while $t=1$ gets you $M=B$. So $t$ represents the amount of displacement along the line, starting from point $A$. Plugging in any other value of $t$ will get you some other point on the line.

So the simple formula for the $3D$ coordinates $(x_M, y_M, z_M)$ of $M$ is $$\left\{ \begin{split} x_M &= x_A + t (x_B-x_A)\\ y_M &= y_A + t (y_B-y_A)\\ z_M &= z_A + t (y_B-z_A)\\ \end{split}\right.$$