Why does the determinant of a Line and a Point results in the direction of the Point to the Line?

Question

I have a point (1, 1) and Line segment that goes from points (1, 2) to (2, 1)

My lecture says that to find the direction of the point to the Line we can create the following Matrix and if the determinant is negative the point is below the line or else it's above. Which works! But why?

Matrix3(
   1, 1, 1, //Point
   1, 2, 1, //Line start
   2, 1, 1  //Line end 
)
//determinant = -1

After some derivation exercises and videos I understand that the determinant of a Matrix gives us the area/volume of the transformation and a negative determinant means that the transformation flipped the vector in the opposite direction.

But why would the determinant also give us the direction of a Point to a Line in the above matrix?

Yep they are. It's in 2D so it's set to 1 so that z direction is ignored. — user438454534, Jul 01 '21 at 13:17
Compare the Formula for distance line-point in 2D with formula for Determinant of 3x3 matrix. — Mauro ALLEGRANZA, Jul 01 '21 at 13:41
https://math.stackexchange.com/questions/488163/meaning-of-signed-volume?rq=1 — Soham, Jul 01 '21 at 13:41
The reason is: vector computation of normal to a line through a point and the sign is accounted for by the direction of the vector. — Mauro ALLEGRANZA, Jul 01 '21 at 13:51
You must have misunderstood your lector: in case you change starting point and endpoint the determinant changes sign, but $(1,1)$ is still "under" the line. What can be said is: if the determinant is negative, the the point is on the right of the line while moving from start to end. Clearly if you change start and end the point appears on the left of the line. — Michael Hoppe, Jul 01 '21 at 13:57

Why does the determinant of a Line and a Point results in the direction of the Point to the Line?

0 Answers0