What is the result of the cross product of two vectors represented in determinants?

The result of the cross product of two vectors can indeed be represented using determinants. Specifically, when calculating the cross product of vectors (\mathbf{a}) and (\mathbf{b}) in three-dimensional space, the formula involves a determinant of a 3x3 matrix formed by the unit vectors ( \mathbf{i}, \mathbf{j}, \mathbf{k} ) and the components of vectors (\mathbf{a}) and (\mathbf{b}).

The determinant approach gives a vector that is orthogonal to both input vectors (\mathbf{a}) and (\mathbf{b}), and its magnitude corresponds to the area of the parallelogram defined by those vectors. The determinant captures both the orientation and the magnitude of this resulting vector, which is precisely why using determinants is pertinent when dealing with cross products.

Other methods mentioned, such as vector addition, dot products, and scalar multiplication, do not yield the same result as the cross product. Vector addition combines the vectors head-to-tail instead of producing another vector perpendicular to the original ones. The dot product results in a scalar that represents the cosine of the angle between the vectors, while scalar multiplication alters the length of a single vector rather than creating a