Possible way of finding a point on a line knowing the vector equations of two lines, a point on one line, and another point.

Question

So basically say you have two lines $l_1$ and $l_2$, and you know their vector equations. And then say you know a point $P$ to be $(x,y,z)$ (a random point in space). You also know that a point $Q(x_1,y_1,z_1)$ lies on say $l_1$. Now you're told that another point $R$ lies on $l_2$, but you don't what that point is, but you are given that $PQ=PR$, so is it possible to find $R$?

Answer 1

You know the point $P$ and the distance $PQ=PR$

You want to find the point $R$ on the line $l_2$.

The point $R$ is on the intersection of a sphere with radius $PQ=PR$ and centered at $P$ with the line $l_2$

Since the equations of such a sphere and the line $l_2$ are known, you solve the simultaneous system to find the possible intersection points.

The system may have two, one, or no solutions depending on the distance $PR$