Answer:
[tex]A=\left[\begin{array}{cc}-\frac{\sqrt{3} }{2} &\frac{1}{2} \\-\frac{1}{2} &-\frac{\sqrt{3} }{2}\end{array}\right][/tex]
Step-by-step explanation:
A counterclockwise rotation through an angle [tex]\theta[/tex], can be described with the help of trigonometric functions.
The matrix: [tex]\left[\begin{array}{cc}\cos \theta&-\sin \theta\\\sin \theta&\cos \theta\end{array}\right][/tex] describes the counterclockwise rotation of the [tex]R^2[/tex] plane by an angle of [tex]\theta[/tex].
For clockwise rotation by an angle of [tex]\theta[/tex] about the origin, we replace [tex]\theta[/tex] by [tex]-\theta[/tex] to obtain:
[tex]\left[\begin{array}{cc}\cos (-\theta)&-\sin (-\theta)\\\sin (-\theta)&\cos (-\theta)\end{array}\right][/tex]
Apply the odd and even properties of the sine and cosine functions to obtain:
[tex]\left[\begin{array}{cc}\cos (\theta)&\sin (\theta)\\-\sin (\theta)&\cos (\theta)\end{array}\right][/tex].
The matrix that describes a rotation of the [tex]R^2[/tex] plane around the origin of [tex]150\degree[/tex] clockwise is:
[tex]\left[\begin{array}{cc}\cos (150\degree)&\sin (150\degree)\\-\sin (150\degree)&\cos (150\degree)\end{array}\right]=\left[\begin{array}{cc}-\frac{\sqrt{3} }{2} &\frac{1}{2} \\-\frac{1}{2} &-\frac{\sqrt{3} }{2}\end{array}\right][/tex].
Therefore the required matrix is:
[tex]A=\left[\begin{array}{cc}-\frac{\sqrt{3} }{2} &\frac{1}{2} \\-\frac{1}{2} &-\frac{\sqrt{3} }{2}\end{array}\right][/tex]
The corresponding linear transformation is:
[tex]T(x,y)=(-\frac{\sqrt{3} }{2}x +\frac{1}{2}y,\frac{1}{2}x- \frac{\sqrt{3} }{2}y)[/tex]
The matrix of the linear transformation from ℝ2 to ℝ2 that rotates any vector through an angle of 150° in the clockwise direction.
is [tex]\left[\begin{array}{ccc}-\sqrt{3}/2 &1/2\\1/2&-\sqrt{3}/2 \\\end{array}\right][/tex].
Matrix for counter-clockwise direction for an angle [tex]\alpha[/tex] is
[tex]\left[\begin{array}{ccc}cos\alpha &-sina\\-sin\alpha &cos\alpha \\\end{array}\right][/tex]
Matrix for clockwise direction for an angle [tex]\alpha[/tex]
Put [tex]\alpha = -\alpha[/tex] in the above matrix
Above matrix becomes
[tex]\left[\begin{array}{ccc}cos\alpha &sina\\sin\alpha &cos\alpha \\\end{array}\right][/tex]
for [tex]\alpha =150[/tex]
Above matrix becomes
[tex]\left[\begin{array}{ccc}cos\ 150 &sin 150\\sin\ 150&cos\ 150 \\\end{array}\right][/tex]
[tex]\left[\begin{array}{ccc}-\sqrt{3}/2 &1/2\\1/2&-\sqrt{3}/2 \\\end{array}\right][/tex]
Therefore, the matrix of the linear transformation from ℝ2 to ℝ2 that rotates any vector through an angle of 150° in the clockwise direction.
is [tex]\left[\begin{array}{ccc}-\sqrt{3}/2 &1/2\\1/2&-\sqrt{3}/2 \\\end{array}\right][/tex].
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