# How do I find a point on a 3D line that has a certain space from a point in space?

suppose I know the coordinates of a point and the equation of a line in space and the coordinates of a point. how do I find a point on this line that has a certain given distance D from that point ?

let's the line equation be

$ax + bx + c = l$

and the point be

$x= p1 , y = p2 , z = p3$

• The step-by-step details of an answer will depend on what kind of "equation of a line in space" you have to start with. You can edit the question to show the form of the "equation"; see math.stackexchange.com/help/notation for guidance on formatting the formulas to make them readable. Another thing: you mention "the coordinates of a point" twice, which might make someone think you have two different points in mind; but I guess you meant just one point, is that right? And is that point on the line somewhere, or just an arbitrary point somewhere in space? – David K Sep 20 '18 at 12:19
• The equation you give represents a plane, not a line. – Aretino Sep 20 '18 at 14:43

Write the equation of the sphere with center $(p_1,p_2,p_3)$ and with radius the given distance $D$: $$(x-p_1)^2+(y-p_2)^2+(y-p_3)^2=D^2.$$ The points you need are the intersections between this sphere and the given line. Their coordinates can be found by solving the system formed by the above equation and the equations describing the line.