What properties does an orthogonal set of vectors have?

An orthogonal set of vectors is characterized by the specific property that each pair of distinct vectors within the set is orthogonal, meaning that their dot product equals zero. This is a fundamental aspect of orthogonality in linear algebra. When vectors are orthogonal, they are at right angles to each other in a geometric sense, which has important implications in various applications, such as in simplifying calculations and decomposing vectors into components.

In mathematical terms, if you have a set of vectors ( v_1, v_2, \ldots, v_n ), they are orthogonal if for any two distinct vectors ( v_i ) and ( v_j ) (where ( i \neq j )), the following condition holds true:

[ v_i \cdot v_j = 0 ]

This shows that distinct vectors in an orthogonal set do not influence each other, making them useful in many linear algebra techniques, such as solving systems of equations and performing projections.

The other properties mentioned in the choices do not encompass the definition of orthogonality. For example, while all vectors in an orthogonal set may indeed have lengths that vary, they do not necessarily have to be of equal length. Additionally, the dot product