Q.Fill in the blanks:

Concept: 3D Coordinate Geometry – Planes and Octants

In three-dimensional space, any two coordinate axes determine a coordinate plane. The $x$-axis and $y$-axis lie in and define the XY-plane (also called the $xy$-plane).

Every point in the XY-plane has zero distance from this plane along the perpendicular direction, which is the $z$-direction. Therefore, all points in the XY-plane have $z$-coordinate equal to zero, giving coordinates of the form $(x, y, 0)$. …

When we move from two dimensions to three, we add a third axis perpendicular to the familiar $xy$-plane. This $z$-axis, together with the $x$ and $y$ axes, creates a framework for locating any point in space using three coordinates $(x, y, z)$.

The three axes naturally define three special planes. Each plane is determined by a pair of axes and represents all points where the third coordinate is zero. These are the fundamental building blocks of 3D coordinate geometry.

Understanding Coordinate Planes

The XY-plane is the flat surface containing both the $x$-axis and the $y$-axis. Every point on this plane has $z = 0$ because it hasn't moved "up" or "down" from the original two-dimensional world. Similarly, the YZ-plane (containing $y$ and $z$ axes) has $x = 0$, and the XZ-plane (containing $x$ and $z$ axes) has $y = 0$.

Now let's fill in each blank systematically:

(i) The plane determined by the $x$-axis and $y$-axis

When two perpendicular lines (axes) lie in the same flat surface, they determine a plane. The $x$-axis and $y$-axis together span the horizontal plane we're most familiar with from 2D geometry.

The $x$-axis and $y$-axis taken together determine a plane known as the XY-plane (or the $xy$-plane, or coordinate plane XY).

(ii) Coordinates of points in the XY-plane

Any point lying in the XY-plane can move freely along the $x$ direction and the $y$ direction, but it cannot have any $z$-component—it must remain flat on that plane. This means the $z$-coordinate is always zero.

The coordinates of points in the XY-plane are of the form $(x, y, 0)$, where $x$ and $y$ can be any real numbers.

(iii) Number of octants created by coordinate planes …