Unit basis vectors emerged from Hamilton’s quaternions, and quite literally form the basis of rotation and attitude. I begin with their role in the dot product, and then study the matrix determinant. This determines the handedness of any three vectors, which is necessary for building a right-handed cartesian coordinate system. That idea naturally gives rise to the cross product, which I study in some detail, including in higher dimensions. The chapter ends with comments on matrix multiplication, and in particular the fast multiplication of sparse 3×3 matrices that we use frequently later in the book.