for our coordinates; then the squared distance to , by Pythagoras, is which can be computed using the matrix multiplication A geometric rotation transforms lines to lines, and preserves ratios of distances between points. From these properties it can be shown that a rotation is a linear transformation of the vectors, and thus can be written in matrix form, . The fact that a rotation preserves, not just ratios, but distances themselves, is stated as or Because this equation holds for all vectors, , one concludes that every rotation matrix, , satisfies the orthogonality condition, Rotations preserve handedness because they cannot change the ordering of the axes, which implies the special matrix condition, Equally important, it can be shown that any matrix satisfying these two conditions acts as a rotation. Multiplication The inverse of a rotation matrix is its transpose, which is also a rotation matrix: The product of two rotation matrices is a rotation matrix: For , multiplication of rotation matrices is generally not commutative. Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the rotation matrices form a group, which for is non-abelian, called a special orthogonal group, and denoted by , , , or , the group of rotation matrices is isomorphic to the group of rotations in an space. This means that multiplication of rotation matrices corresponds to composition of rotations, applied in left-to-right order of their corresponding matrices. Ambiguities The interpretation of a rotation matrix can be subject to many ambiguities. In most cases the effect of the ambiguity is equivalent to the effect of a rotation matrix inversion (for these orthogonal matrices equivalently matrix transpose). Alias or alibi (passive or active) transformation The coordinates of a point may change due to either a rotation of the coordinate system (alias), or a rotation of the point (alibi). In the latter case, the rotation of also produces a rotation of the vector representing . In other words, either and are fixed while rotates (alias), or is fixed while and rotate (alibi). Any given rotation can be legitimately described both ways, as vectors and coordinate systems actually rotate with respect to each other, about the same axis but in opposite directions. Throughout this article, we chose the alibi approach to describe rotations. For instance, represents a counterclockwise rotation of a vector by