The cross product of two vectors, denoted by V × W, produces a third vector (U) that is perpendicular to the two input vectors. This property of being perpendicular, or orthogonal, to both input vectors is closely related to the geometric interpretation of the determinant.
The magnitude of the cross product V × W is given by the formula: |V × W| = |V| * |W| * sin(θ) where |V| is the magnitude of vector V, |W| is the magnitude of vector W, and θ is the angle between vectors V and W.
Now, let's consider the determinant definition of the cross product: V × W = (Vy * Wz - Vz * Wy) * i + (Vz * Wx - Vx * Wz) * j + (Vx * Wy - Vy * Wx) * k
The coefficients of the resulting vector U (i.e., (Vy * Wz - Vz * Wy), (Vz * Wx - Vx * Wz), and (Vx * Wy - Vy * Wx)) can be interpreted as the areas of the parallelograms formed by the projections of V and W onto the three coordinate planes (yz, xz, xy planes, respectively).
The intriguing question here is why the determinant produces orthogonality between the cross product vector U and the original vectors V and W? Is there any simple geometric interpretation that explains this phenomenon, rather than relying on algebraic proofs?
I am seeking a geometric explanation that sheds light on this without resorting to formalistic approaches. Since it's relatively easy to verify algebraically. I have attempted various angles to tackle this, but it still appears somewhat mysterious to me.
