Chapter 11.5, Problem 3E

To determine

To find : the $XY$ coordinates of the point if the coordinate axes are rotated through the indicated angle

Answer to Problem 3E

The coordinates are $\left(X,Y\right)=\left(\sqrt{2},0\right)$

Explanation of Solution

Given information : The angle is $\varphi ={45}^{\circ }$ and the points are $\left(1,1\right)$

Calculation :

The coordinates of $XY$ is obtained as:

  $\begin{array}{l}\left(x,y\right)=\left(1,1\right)\\ \varphi ={45}^{\circ }\\ X=x\cos \varphi +y\sin \varphi \\ X=1\left(\cos {45}^{\circ }\right)+1\left(\sin {45}^{\circ }\right)\\ X=1\left(\frac{1}{\sqrt{2}}\right)+\frac{1}{\sqrt{2}}=\frac{1+1}{\sqrt{2}}\\ X=\frac{2}{\sqrt{2}}\Rightarrow \sqrt{2}\\ Y=-x\sin \varphi +y\cos \varphi \\ Y=-1\left(\sin {45}^{\circ }\right)+1\left(\cos {45}^{\circ }\right)\\ Y=-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}=0\\ \left(X,Y\right)=\left(\sqrt{2},0\right)\end{array}$

Hence,

The coordinates are $\left(X,Y\right)=\left(\sqrt{2},0\right)$