Update

analytic_geometry.md

@@ -8,7 +8,31 @@ Anyway, **how does it work?** Typically we work in a 2D or 3D [Euclidean space](
 
 Analytic geometry is closely related to [linear algebra](linear_algebra.md).
 
-## Example
+## Examples
+
+**Nub example**:
+
+Find the intersection of two lines in 2D: one is a horizontal line with *y* position 2, the other is a 45 degree line going through the [0,0] point in the positive *x* and positive *y* direction, like this:
+
+```
+  y
+  :        _/ line 2
+  :      _/
+_2:_____/_______ line 1
+  :  _/
+  :_/
+--:----------x
+_/:
+  :
+```
+
+The equation of line 1 is just *y = 2* (it consists of all points *[x,2]* where for *x* we can plug in any number to get a valid point on the line).
+
+The equation of line 2 is *x = y* (all points that have the same *x* and *y* coordinate lie on this line).
+
+We find the intersection by finding such point *[x,y]* that satisfies both equations. We can do this by plugging the first equation, *y = 2*, to the second equation, *x = y*, to get the *x* coordinate of the intersection: *x = 2*. By plugging this *x* coordinate to any of the two line equations we also get the *y* coordinate: 2. I.e. the intersection lies at coordinates *[2,2]*.
+
+**Advanced nub example**:
 
 Let's say we want to find, in 2D, where a line *L* intersects a circle *C*. *L* goes through points *A = [-3,0.5]* and *B = [3,2]*. *C* has center at *[0,0]* and radius *r = 2*.