Deducing whether points are in the same plane.

[crude lichen]

I'm trying to decide whether the points down below, given in an ON-base, are in the same plane or not. I tried the following: if the points are called A,B,C,D then they form vectors AB,BC,CD, if they are in the same plane, then these vectors must be linearly dependent, thus the determinant of their coordinates must be 0. $AB = {1,3,1}$, $BC = {1,-4,-2}$, $CD = {0,-3,-3}$. However, I get that their determinant is NOT 0 (I get 7)

[thin heath]

Delusional Pen

[crude lichen]

Is my thinking incorrect? How would onesolve this problem? I think one possible way might be calculating the vector product between two of the vectors above, and obtain the plane's normal equation, and check if the third vector exists within the plane?

[dusty hornet]

Maybe try the cross product between 2 sets of 2 of the vectors and see if their resulting vectors are parallel

[crude lichen]

I'll try that! I'm still wondering why my method above doesn't work though. Pretty sure that three vectors in the same plane means that they are linearly dependent.

[crude lichen]

Thank you, your method worked! I also retried my method, and I realized that I did some stupid mistakes when calculating AB, BC and CD. I now get that the determinant is 0 which implies they are in the same plane, which validates your method too since both agree that they are in the same plane 🙂 !