Chapter 9.3, Problem 34HP

To determine

Whether collinearity and betweenness of points maintained under rotation.

Answer to Problem 34HP

The collinearity and betweenness is maintained under rotation.

Explanation of Solution

To carry out the rotation using matrices the point $\left(x,y\right)$ to be rotated is written as vector.

  $\begin{array}{c}\sqrt{{\left(x^{\prime }\right)}^2+{\left(y^{\prime }\right)}^2}=\sqrt{{\left(x\cos \theta -y\sin \theta \right)}^2+{\left(x\sin \theta +y\cos \theta \right)}^2}\\ =\sqrt{x^2{\cos }^2\theta +y^2{\sin }^2\theta +x^2{\sin }^2\theta +y^2{\cos }^2\theta }\\ =\sqrt{x^2+y^2}\end{array}$

The angle between the vectors.

  $\begin{array}{c}\tan {\theta }^{\prime }=\frac{\frac{x\sin \theta +y\cos \theta }{x\cos \theta -y\sin \theta }-\frac{y}{x}}{1+\left(\frac{x\sin \theta +y\cos \theta }{x\cos \theta -y\sin gq}\right)\left(\frac{y}{x}\right)}\\ =\frac{x^2\sin \theta +y^2\sin \theta }{x^2\cos \theta +y^2\cos \theta }\\ =\frac{\sin \theta }{\cos \theta }\\ =\tan \theta \\ {\theta }^{\prime }=\theta \end{array}$

Hence, the collinerrity and betweenness is maintained under rotation.