Arizona State University (ASU) MAT343 Applied Linear Algebra Exam 2 Practice

Two vectors are considered orthogonal if their dot product is zero. This stems from the geometric interpretation of the dot product, which measures the extent to which two vectors point in the same direction. When the dot product of two vectors is zero, it indicates that the angle between them is 90 degrees, which is the definition of orthogonality.

Orthogonal vectors are significant in various applications, such as in computer graphics, signal processing, and machine learning, as they indicate that the vectors are independent of each other in terms of their directional influence. This property allows for simplifying calculations, particularly in constructing orthonormal bases for vector spaces.

Other options, while they describe relationships between vectors, do not define orthogonality accurately. For example, having equal magnitudes, pointing in opposite directions, or lying on different planes do not necessarily mean that the vectors are orthogonal. Thus, the defining characteristic of orthogonal vectors is indeed their zero dot product, making the second choice the correct answer.