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T2 3 t1 e 1 0 2 2 t2 t1 3 e 2 50 0 2 2 from

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Unformatted text preview: anest way2 to solve this system is to write it as a matrix equation. Using the machinery developed in Section 8.4, we write the above system as the matrix equation AX = X where A= cos(θ) − sin(θ) sin(θ) cos(θ) , X= x y , X= x y Since det(A) = (cos(θ))(cos(θ)) − (− sin(θ))(sin(θ)) = cos2 (θ) + sin2 (θ) = 1, the determinant of A is not zero so A is invertible and X = A−1 X . Using the formula given in Equation 8.2 with det(A) = 1, we find A−1 = cos(θ) sin(θ) − sin(θ) cos(θ) so that X = A− 1 X x y = cos(θ) sin(θ) − sin(θ) cos(θ) x y = x y x cos(θ) + y sin(θ) −x sin(θ) + y cos(θ) From which we get x = x cos(θ) + y sin(θ) and y = −x sin(θ) + y cos(θ). To summarize, Theorem 11.9. Rotation of Axes: Suppose the positive x and y axes are rotated counterclockwise through an angle θ to produce the axes x and y , respectively. Then the coordinates P (x, y ) and P (x , y ) are related by the following systems of equations x = x cos(θ) − y sin(θ) y = x sin(θ) + y cos(θ) and x y = x cos(θ) + y sin(θ) = −x sin(θ) + y cos(θ) We put the formulas in Theorem 11.9 to good use in t...
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