Chapter 11.4, Problem 64E

a.

To determine

To find: The angle between the normal vectors.

Answer to Problem 64E

  $\theta ={65}^{\circ }$

Explanation of Solution

Given information: The figure shows two planes and their normal vectors.

  

Calculation:

The angle between the normal vector $n_1\text{and}{n}_2\text{is}{65}^{\circ }.$

  $\begin{array}{l}{n}_1\Vert \overrightarrow{CE},n_2\Vert \overrightarrow{AC}.\\ \theta =angleACE.\\ mangleACE={90}^{\circ }-mangleBAC\\ mangleACE={90}^{\circ }-({90}^{\circ }-mangleABC).\\ mangleACE={90}^{\circ }-({90}^{\circ }-{65}^{\circ }).\\ \theta ={65}^{\circ }\\ \end{array}$

b.

To determine

To find: An angle between this plane and the blue plane, given a third plane is parallel to $n_1$ , What

Answer to Problem 64E

Thus we got that the angle between the blue plane and the new plane is $\phi ={25}^{\circ }$ .

Explanation of Solution

Given information: The figure shows two planes and their normal vectors.

  

Calculation:

The third plane being parallel to $n_1,\text{}\text{the}\text{normal}{n}_3\text{on}\text{it}\text{is}\text{also}\text{perpendicular}\text{on}{n}_1.$

The angle $\phi$ between $n_1\text{and}{n}_3.$

  $\phi ={25}^{\circ }$

Thus we got that the angle between the blue plane and the new plane is: $\phi ={25}^{\circ }$ .