Question

Find the matrix for the linear transformation which rotates every vector in r2 through an angle of −π/3.

Answer 1

The matrix for the linear transformation for which rotates every vector in r2 is $\left[\begin{array}{cc}\frac{1}{2} &\frac{\sqrt{3} }{2} \\ \frac{\sqrt{3} }{2} &\frac{1}{2} \end{array}\right]$.

In this question,

The angle is -π/3 = -60°

Clockwise rotations are denoted by negative numbers.

Matrix for counter-clockwise direction for an angle α is

$\left[\begin{array}{cc}cos\alpha &-sin\alpha \\-sin\alpha &cos\alpha \end{array}\right]$

Then, matrix for clockwise direction for an angle α

Put α = -α in the above matrix,

⇒ $\left[\begin{array}{cc}cos(-\alpha) &-sin(-\alpha) \\-sin(-\alpha) &cos(-\alpha) \end{array}\right]$

⇒ $\left[\begin{array}{cc}cos\alpha &sin\alpha \\sin\alpha &cos\alpha \end{array}\right]$

Now substitute α = 60°

Then matrix becomes,

⇒ $\left[\begin{array}{cc}cos60 &sin60 \\sin60 &cos60 \end{array}\right]$

⇒ $\left[\begin{array}{cc}\frac{1}{2} &\frac{\sqrt{3} }{2} \\ \frac{\sqrt{3} }{2} &\frac{1}{2} \end{array}\right]$

Hence we can conclude that the matrix for the linear transformation for which rotates every vector in r2 is $\left[\begin{array}{cc}\frac{1}{2} &\frac{\sqrt{3} }{2} \\ \frac{\sqrt{3} }{2} &\frac{1}{2} \end{array}\right]$.

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