Chapter 11, Problem 19CT

To determine

To find angle $\theta$ between two adjacent sides of a tractor fuel tank.

Answer to Problem 19CT

The angle $\theta$ between two adjacent sides of a tractor fuel tank is approximately ${88.45}^{\circ }$ .

Explanation of Solution

Given information:

  

To find the angle between two adjacent sides is given by the formula,

  $\cos \theta =\frac{\left|{\text{n}}_{\text{1}}{\text{×n}}_{\text{2}}\right|}{\Vert {\text{n}}_1\Vert \Vert {\text{n}}_{\text{2}}\Vert }$

  ${\text{n}}_1\text{and}{\text{n}}_{\text{2}}$ are the normal vectors of the planes,

  $\begin{array}{l}{\text{Plane n}}_1\left(10,0,0\right),\left(10,10,0\right),\left(8,8,12\right)\\ {\text{Plane n}}_2\left(0,10,0\right),\left(10,10,0\right),\left(8,8,12\right)\end{array}$

The normal vector ${\text{n}}_1$ is,

  $\begin{array}{c}\text{u}=\left(10-0,10-0,0-0\right)=\left(0,10,0\right)\\ \text{v=}\left(8-10,8-0,12-0\right)=\left(-2,8,12\right)\\ {\text{n}}_1=\text{u}\times \text{v}=\left|\begin{array}{ccc}\text{i}& \text{j}& \text{k}\\ 0& 10& 0\\ -2& 8& 12\end{array}\right|=120\text{i}+20\text{k}\\ {\text{n}}_1=\left(120,0,20\right)\end{array}$

The normal vector ${\text{n}}_2$ is,

  $\begin{array}{c}\text{u}=\left(10-0,10-0,0-0\right)=\left(0,10,0\right)\\ \text{v=}\left(8-0,8-10,12-0\right)=\left(8,-2,12\right)\\ {\text{n}}_2=\text{u}\times \text{v}=\left|\begin{array}{ccc}\text{i}& \text{j}& \text{k}\\ 0& 10& 0\\ -2& 8& 12\end{array}\right|=-120\text{i}-20\text{k}\\ {\text{n}}_2=\left(0,-120,-20\right)\end{array}$

By using $\cos \theta =\frac{\left|{\text{n}}_{\text{1}}{\text{×n}}_{\text{2}}\right|}{\Vert {\text{n}}_1\Vert \Vert {\text{n}}_{\text{2}}\Vert }$ ,

  $\begin{array}{c}\cos \theta =\frac{\left|{\text{n}}_{\text{1}}{\text{×n}}_{\text{2}}\right|}{\Vert {\text{n}}_1\Vert \Vert {\text{n}}_{\text{2}}\Vert }\\ \cos \theta =\frac{\left|\left(120,0,20\right)\text{×}\left(0,-120,-20\right)\right|}{\Vert \left(120,0,20\right)\Vert \Vert \left(0,-120,-20\right)\Vert }\\ \cos \theta =\frac{20\left|6\left(0\right)+0\left(6\right)+1\left(1\right)\right|}{20\sqrt{6^2+0^2+1^2}\sqrt{0^2+6^2+1^2}}\\ \cos \theta =\frac{1}{\sqrt{37}\sqrt{37}}\\ \cos \theta =\frac{1}{37}\\ \theta \approx {88.45}^{\circ }\end{array}$

Therefore we can conclude that the angle $\theta$ between two adjacent sides of a tractor fuel tank is approximately ${88.45}^{\circ }$ .