10.7 Two rotations. Two 3-vectors x and y are related as follows. First, the vector x...

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10.7 Two rotations. Two 3-vectors x and y are related as follows. First, the vector x is rotated 40° around the e3 axis, counterclockwise (from e₁ toward e2), to obtain the 3-vector z. Then, z is rotated 20° around the e₁ axis, counterclockwise (from e2 toward e3), to form y. Find the 3 x 3 matrix A for which y = Ax. Verify that A is an orthogonal matrix. Hint. Express A as a product of two matrices, which carry out the two rotations described above. 10.7 Two rotations. Two 3-vectors x and y are related as follows. First, the vector x is rotated 40° around the e3 axis, counterclockwise (from e₁ toward e2), to obtain the 3-vector z. Then, z is rotated 20° around the e₁ axis, counterclockwise (from e2 toward e3), to form y. Find the 3 x 3 matrix A for which y = Ax. Verify that A is an orthogonal matrix. Hint. Express A as a product of two matrices, which carry out the two rotations described above.