# Determinant – Find the corner product of two multivariate vectors and derive the meaning

I was trying to solve:
$$-2dx_ {1} land dx_ {4} ( begin {bmatrix} 2 \ 3 \ -5 \ 2 end {bmatrix}, begin {bmatrix} 2 \ 3 \ 4 \ -5 end {bmatrix}$$
I solved this problem by removing the scalar and then finding the determinant of the matrix given by (let $$u$$ to be the first column vector and $$v$$ to be the following):
$$det begin {bmatrix} -2 * dx_1 (u) & -2 * dx_1 (v) \ dx_4 (u) & dx_4 (v) end {bmatrix} = – 2 * det begin {bmatrix} dx_1 (u) & dx_1 (v) \ dx_4 (u) & dx_4 (v) end {bmatrix} = – 2 * det begin {bmatrix} 2 & 2 \ 2 & -5 end {bmatrix} = – 2 (-14) = 28$$
We must then explain what this number represents and that is where my confusion influences. I understand that this represents the surface of the parallelogram formed from $$u$$ and $$v$$ being projected on the $$partial_ {x_1} partial_ {x_4}$$ plane. Should this area be positive or negative? Without the scalar $$-2$$ the oriented area would go back since $$u$$ at $$v$$ would be clockwise, but with the scalar, it would be in the opposite direction of the hands of a watch not to change the sign? I guess my main confusion comes from whether or not the determinant has integrated orientation and that the scalar of one of the forms-one affects the orientation?