Chapter 10.4, Problem 66E

(a)

To determine

To find: The angle between two normal vectors.

Answer to Problem 66E

The angle between two normal vectors $\cos \theta =\frac{n_1\cdot n_2}{\left|n_1\right|\left|n_2\right|}$ .

Explanation of Solution

Given information:

The given figure is,

  

Calculation:

Consider the figure.

The angle between two planes is generally calculated with the knowledge of angle their normal.

The angle between two planes is the angle between two normal. $n_1$ and $n_2$ are two normal, the angle between two normal can be,

  $\cos \theta =\frac{n_1\cdot n_2}{\left|n_1\right|\left|n_2\right|}$ .

Therefore, the angle between two normal vectors $\cos \theta =\frac{n_1\cdot n_2}{\left|n_1\right|\left|n_2\right|}$ .

(b)

To determine

To find: The angle between this plane and the blue plane when the third plane is parallel to $n_1$ .

Answer to Problem 66E

The angle between this plane and the blue plane is $n_{1}b=0$ .

Explanation of Solution

Given information:

The given figure is,

  

Calculation:

Consider the figure.

If, the third plane is parallel to $n_1$ . The angle can be,

  $n_{1}b=0$

Therefore, the angle between this plane and the blue plane is $n_{1}b=0$ .

(c)

To determine

To find: The angle between this plane and the blue plane when the third plane is parallel to $n_2$ .

Answer to Problem 66E

The angle between this plane and the blue plane is $n_{2}b=0$ .

Explanation of Solution

Given information:

The given figure is,

  

Calculation:

Consider the figure.

If, the third plane is parallel to $n_2$ . The angle can be,

  $n_{2}b=0$

Therefore, the angle between this plane and the blue plane is $n_{2}b=0$ .