Calculating incremental coordinate change along a 3D vector

Question

This is pretty trivial, but I've not managed to resolve it to my satisfaction and it has reached the stage where fresh eyes are definitely useful!

I have an xyz point, and a 3D vector originating at that point. I would like to be able to shift one of the coordinates by a fixed amount, and calculate the change in the other two coordinates as I traverse the vector.

For example:

Starting at the xyz point (24.2778,30.0526,105.738) I have a vector with components (15.7618,19.4226,68.7535).

I would like to move along the vector 2.60662 in z, and then calculate the corresponding points for x and y (still sitting along this original vector).

Is it correct to calculate the unit vector in each direction, and then simply do:

new x = original_x + (x_unit_vector * 2.60662)

new y = original_y + (y_unit_vector * 2.60662)

Doing this yields a new xyz point at (24.8393, 30.7446, 108.188) which seems reasonable, but I'd like to be more confident in my approach than that!

Many thanks.

Answer 1

Suppose you have a point $P$, and you have a 3d vector pointing from it denoted by $N$. Then the equation for the line represented by those two components is $X=P+Nt$, where t is some number. And so you have $$X_x=P_x+N_x t\\X_y=P_y+N_y t\\X_z=P_z+N_z t$$ When $t=0$, then you have the location of your initial point. Now you have shifted one of these values, let's say the $P_x$ value by $n$, then your new point is going to be at $P_x+n$. So now you have $$P_x+n=P_x+N_xt\\t=n/N_x\\X_y =P_y+N_y(n/N_x)\\X_z =P_z+N_z(n/N_x)$$The way you wrote it doesn't necessarily work because you are implying that to get the new $z$ coordinate, all you have to do is do the new z = original_z + (z_unit_vector * 2.60662), which isn't true.