Let be a vector pointing from the fixed point O of a rigid body to an arbitrary point P of the body. (iii) The green cross product in x-y plane is perpendicular to the screen, pointing away from the reader. (ii) The blue vectors are obtained by rotation into the screen. (i) The red vectors and black axis n̂ are in the plane of the screen. Explicit expression of rotation operator Likewise the determinant is −1 for an improper rotation. It was just shown that for a proper rotation the columns of R are orthonormal and satisfy, The determinant of a 3×3 matrix with column vectors a,ī, and c can be written as scalar triple product. This was proved above, an alternative proof is the following: The determinant of a proper rotation matrix is 1 and of an improper The last two equations can be condensed into one equation Its column vectors form a left-handed set, i.e., The matrix R is an improper rotation matrix if The matrix R is a proper rotation matrix, if it is Indeed, invoking some properties of determinants, one can proveĪ compact way of presenting the same results is the following. Orthogonal matrices come in two flavors: proper (det = 1) and improper (det = −1) rotations. Writing out matrix products it follows that both the rows and the columns of the matrix are orthonormal (normalized and orthogonal). Instead, the rigid body could have been left invariant and the Cartesian frame could have been rotated, this also leads to new column vectors of the form p′ ≡ R p, such rotations are referred to as passive. Note that the Cartesian frame is fixed here and that points of the body are rotated, this is known as an active rotation. Thus, an orthogonal matrix leads to a unique rotation. If we map all points P of the body by the same matrix R in this manner, we have rotated the body. Multiply p by the orthogonal matrix R, then p′ = R p represents the rotated point P′ (or, more precisely, the vector is represented by column vector p′ with respect to the same Cartesian frame).
Expressing this vector with respect to a Cartesian frame in O gives the column vector p (three stacked real numbers). So, a rotation gives rise to a unique orthogonal matrix.Ĭonversely, consider an arbitrary point P in the body and let the vector connect the fixed point O with P.
Since this holds for any pair a and b it follows that a rotation matrix satisfiesįor finite-dimensional matrices one shows easilyĪ matrix with this property is called orthogonal. The invariance of the inner product under the rotation operator leads to First we define column vectors (stacked triplets of real numbers given in bold face):Īnd observe that the inner product becomes by virtue of the orthonormality of the basis vectors It is easily shown that a similar vector-matrix relation holds. If the body is 3-dimensional-it contains three linearly independent vectors with origins in the invariant point-it holds that for any pair of vectors and in ℝ 3 the inner product is invariant, that is,Ī linear map with this property is called orthogonal. Given a basis of the linear space ℝ 3, the association between a linear map and its matrix is one-to-one.Ī rotation (for convenience sake the rotation axis and angle are suppressed in the notation) leaves the shape of a rotated rigid body intact, so that all distances within the body are invariant. In a more condensed notation this equation can be written as
Write for the unit vector along the rotation axis and φ for the angle over which the body is rotated, then the rotation operator on ℝ 3 is written as ℛ( φ, n̂).Įrect three Cartesian coordinate axes with the origin in the fixed point O and take unit vectors along the axes, then the 3×3 rotation matrix is defined by its elements By Euler's theorem follows that then not only the point is fixed but also an axis-the rotation axis- through the fixed point. By a translation all points of the body are displaced, while under a rotation at least one point of the body stays in place. In general a motion of a rigid body (which is equivalent to an angle and distance preserving transformation of affine space) can be described as a translation of the body followed by a rotation. Connection of an orthogonal matrix to a rotation 5.1 Case that "from" and "to" vectors are anti-parallel.4 Explicit expression of rotation matrix.3 Explicit expression of rotation operator.1 Connection of an orthogonal matrix to a rotation.