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The point $(1,-1,5)$ obviously satisfies the first equation, and the lines aren’t parallel, so if you’re not getting that as the point of intersection you’re making a mistake somewhere along the way.
Given two lines joining A,B and C, D in 3D how do I figure out if they intersect, where they intersect and the ratio along AB at which the intersection happens? I can quite hapilly work out the equation for the lines in different forms. I'm guessing that you need to change them to parametric form, equate the equations and do some algebra ...
Find a line parallel to two planes and intersecting two lines. 0. Vectors and directed line segments.
And you only need to prove the existence of two intersecting lines. That means: Choose any two points, draw the line through the line segment determined by those two points, and then draw a third point not on the first line. The only line through that third point that won't intersect the first line would be a line parallel to the first line.
The dark black lines are the original lines, and the light gray lines are offset by one radius. Where they intersect is the center of the corner radius (red). Here is a graphical representation of some C# code I did to implement this algorithmically.
Those two lines are nonparallel and they do not intersect (I checked that). Using the vector product I computed the normal (the line orthogonal to both of these lines), and the normal is $(3, -2, 1)$. Now I have the direction vector of the line which will intersect both of my nonparallel lines.
$\begingroup$ if these two lines are intersect they have a intersection point that point lies on both lines so you can get that point as $<a,b,c>$ then tried to get system of equation and solve it $\endgroup$
From the coefficients of x, y and z of the general form equations, the first plane has normal vector $\begin{pmatrix}1\\2\\1\end{pmatrix}$ and the second has normal vector $\begin{pmatrix}2\\3\\-2\end{pmatrix}$, so the line of intersection must be orthogonal to both of these.
Solution for finding intersection of two lines described by parametric equation. 3.
In the examples given, the lines are defined by two points, but a direction vector may be obtained by taking the difference in the coordinates of the two given points. Here is the construction. We leave the proof as an exercise, so that the interested reader may benefit by working it out (hint: the Law of Sines is helpful).