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4_rotations - 4 Orthogonal transformations and Rotations A...

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4. Orthogonal transformations and Rotations A matrix is defined to be orthogonal if the entries are real and (1) A 0 A = I . Condition (1) says that the gram matrix of the sequence of vectors formed by the columns of A is the identity, so the columns are an orthonormal frame. An orthog- onal matrix defines an orthogonal transformation by mutiplying column vectors on the left. Condition (1) also shows that A is a rigid motion preserving angles and distances. A matrix satisfying (1) preserves the dot product; the dot product of two vectors is the same before and after an orthogonal transformation. This can be written as Av · Aw = v · w for all vectors v and w . This is true by the definition (1) of orthogonal matrix since Av · Aw = ( Av ) 0 Aw = v 0 A 0 Aw = v 0 Iw = v 0 w = v · w . Thus lengths and angles are preserved, since they can be written in terms of dot products. The orthogonal transformation are a group since we can multiply two of them and get an orthogonal transformation. This is because if A and B are orthogonal, then A 0 A = I and B 0 B = I . So ( AB ) 0 AB = B 0 A 0 AB = I , showing that AB is also orthogonal. Likewise we can take the inverse of an orthog- onal transformation to get an orthogonal transformation. Orthogonal transformations have determinant 1 or - 1 since by (1) and properties of determinant, (det A ) 2 = det( A 0 ) det A = det( A 0 A ) = det I = 1 . 4.1. The rotation group. Orthogonal transformations with determinant 1 are called rotations, since they have a fixed axis. This is discussed in more detail below. The rotations also form a group. If we think of an orthogonal matrix A as a frame A = ( v 1 , v 2 , v 3 ) , then the determinant is the scalar triple product v 1 · ( v 2 × v 3 ) . The frame is right handed if the triple product is 1 and left handed if it is -1. The frame is the image of the right handed standard frame ( e 1 , e 2 , e 3 ) = I under the transformation A . Thus A preserves orientation (right-handedness) if the determinant is 1. 1
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2 4.1.1. Rotations and cross products. Rotations are orthogonal transformations which preserve orientation. This is equivalent to the fact that they preserve the vector cross product: (2) A ( v × w ) = Av × Aw , for all rotations A and vectors v and w . Recall the right hand rule in the definition
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