er the direction cosine matrix, C = [C], between two sets of right- hand orthogonal unit...

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er the direction cosine matrix, C = [C], between two sets of right- hand orthogonal unit vectors (, , ds) and (b, 5, 53), defined as ba [6 || ARBON (c) Show that C| = 1. (a) Show that the direction cosine matrix C is an orthonormal matrix; i.e.. CC =1=CC. cc. I Hint: [b b b3] = [, z s]C and [C C12 C13 U22 63] LC31 C23 02 C32 C33 L3 P b ba CATATATATATA (b) Show that the ijth element of the direction cosine matrix C is equal to the ijth cofactor of C; i.e., C; = (-1) Mj or adj C = [(1)+M] [C] = C. Banne Hint: Using b, b 53, show that C = C2C33 - C23 C32 C12=C23C31 - C21 C33 C13=C21C32 - C22 C31 0 :: 1 Hint: C = adj C/ICI. (d) Find six independent equations for C, using the row-orthonormality condition, CC (e) Find six independent equations for C, using the column-orthonormality condition, CC - 1. Are these six equations independent of those of part (d)? (f) Finally, show that only three of the nine direction cosines are independent. Note: Three direction cosines, however, do not uniquely define the orientation of two reference frames. er the direction cosine matrix, C = [C], between two sets of right- hand orthogonal unit vectors (, , ds) and (b, 5, 53), defined as ba [6 || ARBON (c) Show that C| = 1. (a) Show that the direction cosine matrix C is an orthonormal matrix; i.e.. CC =1=CC. cc. I Hint: [b b b3] = [, z s]C and [C C12 C13 U22 63] LC31 C23 02 C32 C33 L3 P b ba CATATATATATA (b) Show that the ijth element of the direction cosine matrix C is equal to the ijth cofactor of C; i.e., C; = (-1) Mj or adj C = [(1)+M] [C] = C. Banne Hint: Using b, b 53, show that C = C2C33 - C23 C32 C12=C23C31 - C21 C33 C13=C21C32 - C22 C31 0 :: 1 Hint: C = adj C/ICI. (d) Find six independent equations for C, using the row-orthonormality condition, CC (e) Find six independent equations for C, using the column-orthonormality condition, CC - 1. Are these six equations independent of those of part (d)? (f) Finally, show that only three of the nine direction cosines are independent. Note: Three direction cosines, however, do not uniquely define the orientation of two reference frames.