Q.The three coordinate planes divide the space into ________ parts.

The key idea here is the division of 3D space by coordinate planes, leading to what are known as octants.

To understand this, consider the progression from lower dimensions:

1. In one dimension, a single point divides a line into $2$ parts.
2. In two dimensions, two perpendicular lines (the $x$ and $y$ axes) divide a plane into $2 \times 2 = 4$ parts, called quadrants. …

When we talk about dividing space, it's helpful to start with simpler dimensions to build intuition.

Consider a 1-dimensional line. If you place a single point on this line, it divides the line into two parts: one to the left and one to the right of the point.

Now, move to a 2-dimensional plane. If you draw a single line on this plane, it divides the plane into two parts. If you draw a second line that intersects the first, these two lines divide the plane into four distinct regions. These are what we call quadrants in a Cartesian coordinate system, defined by the $x$-axis and $y$-axis. Each quadrant is characterized by the signs of the $x$ and $y$ coordinates (e.g., $x>0, y>0$ for the first quadrant).

Extending this idea to 3-dimensional space, we have three coordinate planes. Each plane acts as a "divider," similar to how a line divides a 2D plane. Since these three planes are mutually perpendicular and intersect at a single point (the origin), they carve up the space into multiple distinct sections.

Here's how we determine the number of parts:

1. Identify the Coordinate Planes: In a 3D Cartesian coordinate system, we have three primary planes:

   • The $xy$-plane, where $z=0$.
   • The $yz$-plane, where $x=0$.
   • The $zx$-plane, where $y=0$.

2. Division by Each Plane:

   • The $xy$-plane ($z=0$) divides the space into two halves: one where $z>0$ (above the plane) and one where $z<0$ (below the plane).
   • Similarly, the $yz$-plane ($x=0$) divides the space into two halves: one where $x>0$ and one where $x<0$.
   • And the $zx$-plane ($y=0$) divides the space into two halves: one where $y>0$ and one where $y<0$. …