The 2x2 matrix can rotate the point [x, y] in the counterclockwise direction by

Question

The 2x2 matrix can rotate the point [x, y] in the counterclockwise direction by an angle of theta about the origin. What is the determinant off this matrix?. What is its inverse? Using the matrix in 9), rotate the parabola y = 2x2 +3 by 30 degree. State the equation of the rotated parabola and then state the coordinates of the rotated vertex and the points (-1, 5) and (1, 5). Hint: Write the parabola as a vector in parametric form.

Explanation / Answer

9)Determinant = cos(teta)^2+sin(teta)^2 = 1 Inverse: [cos(teta) sin(teta)] [-sin(teta) cos(teta) 10)X = 0.866025404x-0.5y Y = 0.5x+0.866025404y =>0.5X = (0.866025404*0.5)x-(0.5*0.5)y 0.866025404Y = (0.5*0.866025404)x+(0.866025404*0.866025404)y =>(0.866025404*0.866025404)+(0.5*0.5) * y = 0.866025404Y-0.5X =>y = 0.866025404Y-0.5X and 0.866025404x = X +0.5y = X+(0.5*0.866025404)Y-(0.5*0.5)X =>x = 0.866025404X+0.5Y y = 2x^2+3 =>(0.866025404Y-0.5X)^2 = 2(0.866025404X+0.5Y)^2+3 =>0.75Y^2+0.25X^2-0.866025404XY = 1.5X^2+0.5Y^2+1.732050808XY+3 =>0.25Y^2 = 1.25X^2+2.598076211XY+3 =>Y^2 = 5X^2 + 10.39230485XY + 12 (-1,5) => x = -1, y = 5 =>X = 0.866025404x-0.5y = -3.366025404, Y = 0.5x+0.866025404y = 3.83012702 (1,5) => x = 1, y = 5 =>X = 0.866025404x-0.5y = -1.633974596, Y = 0.5x+0.866025404y = 4.83012702

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