Chapter 11.4, Problem 39E

To determine

Tofind: whether the planes are parallel, perpendicular, or neither. If they are neither parallel nor perpendicular, find the angle of intersection.

Answer to Problem 39E

Given planes are perpendicular.

Explanation of Solution

Given planes: $2x-z=1$ and $4x+y+8z=10$

Calculation:

The direction ratios of normal to given planes are $\langle 2,0,-1\rangle$ and $\langle 4,1,8\rangle$ .

We know that if the direction ratios of normal to the two planes are $\langle a_1,b_1,c_1\rangle$ and $\langle a_2,b_2,c_2\rangle$ , then the planes are perpendicular if

  $a_1{a}_2+b_1{b}_2+c_1{c}_2=0$

And the planes are parallel if

  $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

Here,

  $a_1{a}_2+b_1{b}_2+c_1{c}_2=2\left(4\right)+\left(0\right)\left(1\right)+\left(-1\right)\left(8\right)=0$

Therefore, the given planes are perpendicular.