Chapter 10.1, Problem 29E

To determine

Tofind:The octants in which the point $\left(x,y,z\right)$ is located such that $xy<0$ .

Answer to Problem 29E

The point $\left(x,y,z\right)$ is located in the any of the second, fourth, sixth, eighth octants such that $xy<0$ .

Explanation of Solution

Given information:

The condition is $xy<0$ .

Calculation:

According to the condition $xy<0$ two cases arise:

First case when $x<0$ and $y>0$ . Then the point is behind the $yz$ -plane and also it is to the right of $xz$ -plane. So, the point $\left(x,y,z\right)$ can be in any of the second and sixth octant.

Second case is when $x>0$ and $y<0$ .Then the point is in front of $yz$ -plane and also it is to the left of $xz$ -plane. So, the point $\left(x,y,z\right)$ can be in any of the fourth and eighth octant.

Now combine all the octants. The point $\left(x,y,z\right)$ can be in any of the second, fourth, sixth, eighth octants.

Therefore, the point $\left(x,y,z\right)$ is located in the any of the second, fourth, sixth, eighth octants such that $xy<0$ .