You need to more precise in your question. There is no guarantee two lines interection in 3-d space. If they do intersect, they must lie in the same plane. And that plane would only need to be 2-D, aside from the trivial case where the lines are the same.
If the two lines do intersect then x value for the two lines will be equal, as well as the y value & z value, in the equations of the two lines.
Using parametric equations, If line1 is given by: x = x1 + a*t, y = y1 + b*t, and z = z1 + c*t; and line 2 is: x = x2 + d*s, y = y2 + e*s, and z = z2 + f*s, where x1,y1,z1 & x2,y2,z2 are starting points [a,b,c] & [d,e,f] are direction vectors, and s & t are the parameters.
Since the x values must be equal, set x1+a*t = x2+d*s, and y1+b*t = y2+e*s, then you have two equations and two unknowns. Solve for t and s, then substitute into the z equations to find the z coordinate (they will both come up with the same z-value if indeed the lines do intersect). Even if both equations use 't' as the parameter, you need to treat them as two independent variables(so change one of them to s), since the lines could be changing differently as the parameter variable changes.
a point
Parallel lines.
A coordinate plane.
Yes, they are.
yes... In fact, any two lines on the same plane that are not parallel will share a point of intersection.
Two parallel lines, a plane and a line in a plane parallel to it.
a point
Origin
Parallel lines.
A coordinate plane.
A right angle.
The intersection of two lines is always a point or the line itself. The intersection of a line with plane also the same as above.
Yes, they are.
yes... In fact, any two lines on the same plane that are not parallel will share a point of intersection.
coordinate plane
point I believe the word you're looking for is "intersection". Two non-parallel lines that lie in the same plane will have one point in common where they cross, and that point is the intersection.
Theorem: If two lines intersect, then exactly one plane contains both lines. So, when two or more lines intersect at one point, they lie exactly in the same plane. When two or more lines intersect at one point, their point of intersection satisfies all equations of those lines. In other words, the equations of these lines have the same solution, which is the point of intersection.