Why is the cross product of two vectors, ixj, equal to zero?

The statement regarding the cross product of two vectors, specifically the cross product of the unit vectors i and j, yielding a result of zero is not aligned with the correct principles of vector operations.

The cross product of two vectors yields a vector that is perpendicular to the plane defined by the two original vectors, and its magnitude is given by the product of the magnitudes of the two vectors and the sine of the angle between them. In the case of unit vectors i and j, which are orthogonal (separated by 90 degrees), their cross product is actually equal to k, the unit vector in the z-direction—not zero.

If we were to consider two vectors that do result in a zero cross product, it would typically occur when they are parallel or anti-parallel. Parallel vectors point in the same or opposite directions, leading to an angle of 0 degrees or 180 degrees between them, for which the sine of the angle is zero. This results in the cross product also being zero.

In summary, the correct understanding of the area covered by the cross product stems from recognizing the relationship between the direction and the angle between the vectors being analyzed. In any situation where two vectors are parallel, the cross product will indeed yield a vector