Proving that area of a triangle can be expressed with the determinant

Question

I know that this might be a easy problem, but I'm stuck. I don't know how I'm supposed to use the information given in the task (the vectors)to prove that the area is equal to 1/2 det(P). I thought about starting with explaining why det(P) is equal to the area of a parallelogram and so divide by two. But still, I'm not sure how I'm supposed to use the vectors to do this. Really appreciate some help!

It is standard that $|\det P[$ is the area of the parallelogram built on $P_0, P_1$ and $P_2$ — Bernard, Mar 11 '21 at 23:40

score 1 · Answer 1 · answered Mar 12 '21 at 00:01

1

Hint:

develop the determinant and see that it coincides with $r1 \times r_2$.

answered Mar 12 '21 at 00:01

G Cab

35,272

score 1 · Answer 2 · answered Mar 12 '21 at 00:14

1

This image should help to show the 'why' connecting the row-vectors and the determinant:

https://i.stack.imgur.com/a7QGB.png

answered Mar 12 '21 at 00:14

Carson Bentley

11

1

Found a more formal proof: https://textbooks.math.gatech.edu/ila/determinants-volumes.html – Carson Bentley Mar 12 '21 at 00:44

Proving that area of a triangle can be expressed with the determinant

2 Answers2