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3_frames

# 3_frames - 3 Frames In 3D space a sequence of 3 linearly...

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3. Frames In 3D space, a sequence of 3 linearly independent vectors v 1 , v 2 , v 3 is called a frame, since it gives a coordinate system (a frame of reference). Any vector v can be written as a linear combination v = x v 1 + y v 2 + z v 3 of vectors in the frame and x, y, z are called the coordinates of v in this frame. In computations involving three dimensional space it is often necessary to find the coordinates of vectors in many different frames. When we have two frames there are two different sets of coordinates for each point and there is a matrix to change coordinates between the frames. Here we discuss some of the basic theory of frames and coordinate changes. A frame called the Frenet frame is useful in the study of curves. It is called a moving frame because there is one at each point on the curve, and the points are considered as a function of a parameter t often thought of as time. The idea of a Frenet Frame can be adapted to study the shape of of long molecules such as DNA and proteins, as will be discussed in a later chapter. 3.1. Basic definitions. A sequence of three vectors v 1 , v 2 , v 3 can be made into the columns of a 3 × 3 matrix denoted ( v 1 , v 2 , v 3 ). If the determinant of this matrix is not 0, then the sequence of vectors is called a frame and the vectors are linearly independent . We will not distinguish between the frame and the matrix. The vectors e 1 = 1 0 0 e 2 = 0 1 0 e 3 = 0 0 1 form a frame ( e 1 , e 2 , e 3 ) which is the identity matrix I . This is called the standard basis or lab frame . The determinant of the matrix ( v 1 , v 2 , v 3 ) is equal to the scalar triple product , v 1 · ( v 2 × v 3 ) = det( v 1 , v 2 , v 3 ) . If the determinant is positive, the frame is said to be right-handed , and if the determinant is negative, the frame is said to be left-handed . 3.2. Frames and gram matrices. If F = ( v 1 , v 2 , v 3 ) is a matrix, the entry in row i , column j of the matrix F t F is the dot product v i · v j . This matrix is called the gram matrix of the vectors v 1 , v 2 , v 3 . The frame is said to be an orthogonal frame if the vectors are mutually perpen- dicular. If all of the vectors are of length 1 in an orthogonal frame, it is called an orthonormal frame . The condition that F is an orthonormal frame can be written as v i · v j = ( 1 if i = j 0 if i 6 = j.

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