Final answer:
The product of two 3x3 matrices equals zero when one of them is a zero matrix or when their rows and columns have orthogonality akin to unit vectors on a Cartesian plane, resulting in zero dot products.
Step-by-step explanation:
The product of two 3x3 matrices equals zero when either one of the matrices is a zero matrix, or when the matrices contain rows and columns that result in the dot product of these rows and columns being zero. In a Cartesian coordinate system, we observe this phenomenon when dealing with unit vectors of axes that are orthogonal to each other, leading to a dot product of zero since the cosine of 90° is zero. Similarly, a zero vector product also occurs when we have vectors that are either parallel or antiparallel since the sine of 0° or 180° is zero, and the vector product depends on the sine of the angle between vectors.