I have been reading internet and tried to understand the explanation of the sign of a determinant of a $2\times 2$ matrix.
if I have a matrix
\begin{array}{cc} a & b \\ c & d\\ \end{array}
the determinant is : ad-cb. I read on the wikipedia that if the angle between the first and second vector turns in clockwise direction, it is negative. this is my example:
i want the angle that i draw. I have calculated the 2 vectors:
vect1=(-28, 1)
vect2=(7, 102)
and the det(vect2, vect1) I have positive. I don't understand why it is positive while the angle turns in clockwise direction.
Can someone explain me what the sign of the determinant means really?
Graphing your vectors on Wolfram Alpha
you'll see that you've graphed the second vector incorrectly. Using the image here, you'll see that the angle from vector $2$ to vector $1$ is indeed counter-clockwise, and so the value of the determinant you computed should be positive.
Otherwise, if you need for the second vector to point downward, then you should use $[7, -102]$, which will then yield a determinant that is negative.
Remark: If this question arose in the context of a computing environment, then the usual "rules" that apply to the Cartesian Coordinate system (or here, in what applies for the sign of the determinant of two vectors in $\mathbb R^2$) will not apply when the y-axis is taken to be at the top (of the screen), such that numbers become increasingly large as you descend to the bottom of the screen. In this context, a clockwise rotation from vector $2$ to vector $1$ should be positive.