Question

Consider two rectangular Cartesian coordinate systems that are translated and rotated with respect to each other. The transformation between the two coordinate systems is given by x? = c + Lx where c is a vector and L = [lij] is the matrix of directional cosines, lij ? e?i · e?j Deduce that the following orthogonality conditions hold: L · LT = I or liklkj = ?ij. That is, L is an orthogonal matrix.

          Consider two rectangular Cartesian coordinate systems that are translated and rotated with respect to each other. The transformation between the two coordinate systems is given by

x? = c + Lx

where c is a vector and L = [lij] is the matrix of directional cosines,
lij ? e?i · e?j
Deduce that the following orthogonality conditions hold:
L · LT = I or liklkj = ?ij.
That is, L is an orthogonal matrix.

Consider two rectangular Cartesian coordinate systems that are translated and rotated with respect to each other. The transformation between the two coordinate systems is given by

x? = c + Lx

where c is a vector and L = [lij] is the matrix of directional cosines,
lij ? e?i · e?j
Deduce that the following orthogonality conditions hold:
L · LT = I or liklkj = ?ij.
That is, L is an orthogonal matrix.

Added by Leslie R.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Adi S

10/27/2022

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Consider two rectangular Cartesian coordinate systems that are translated and rotated with respect to each other. The transformation between the two coordinate systems is given by x̄ = c + Lx where c is a vector and L = [lij] is the matrix of directional cosines, lij ≡ êi · êj Deduce that the following orthogonality conditions hold: L · LT = I or liklkj = δij. That is, L is an orthogonal matrix.