## What is the XZ plane?

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.

Now, we are going to extend this for space. In space, we need three numbers for plotting. These three numbers can be represented as an ordered triple (a, b, c) where a, b and c are real numbers and a is the x-coordinate, b is the y-coordinate, and c is the z-coordinate. We call this a three-dimensional space, and it contains three perpendicular axes: x-axis, y-axis, and z-axis.

## Three-Dimensional Coordinate Space

In three-dimensional coordinate space, let us see what the three coordinate axes determine. We can draw a three-dimensional space, by fixing O as the origin and the x-axis, the y-axis and the z-axis passing through O that are perpendicular to each other. We usually assume that the x-axis and the y-axis are horizontal and the z-axis is vertical. From the figure below, it is clear that all these three axes intersect at O.

These three coordinate axes determine three mutually perpendicular planes, the XY plane, the YZ plane, and the XZ plane, which are sometimes called the coordinate planes. From the figure below, we can say that the XY plane contains the x-axis and the y-axis, the YZ plane contains the y-axis and the z-axis, and the XZ plane contains the x-axis and the z-axis.

## Visualization of XZ plane

In this description, we are going to visualize the three-dimensional space, because it seems to be difficult to imagine the three-dimensional space. Before visualizing we will discuss, how space is divided into eight parts. From the above figure, it is clear that the three coordinate planes, the XY plane, the YZ plane, and the XZ plane, divide the whole space into eight compartments which are known as eight octants. The sign of the coordinates of a point depends upon the octant in which the point lies.

It is very difficult for us to imagine the figure of three-dimensional space. So, let us consider the left wall as the XZ plane, the right wall as the YZ plane, and the floor as the XY plane. From the figure below, we can say that the x-axis moves along the intersection of the floor and the left wall, the y-axis moves along the intersection of the floor and the right wall, and the z-axis moves up from the floor toward the ceiling along the intersection of the right and left walls. All three axes extend in both positive as well as negative directions. You can imagine the eight octants connected by the same corner O. In three dimension method, the XZ plane represents the width and height of an object.

## Projection

Let P be any point in the space and we can label P as (a, b, c) where a, b, c are real numbers and a is the x-coordinate, b is the y- coordinate and c is the z-coordinate. The numbers a, b, c are also called the coordinates of P. If we draw a perpendicular line from P towards the XZ plane, we get a point Q and the coordinates of Q is (a, 0, c). We call this point Q(a, 0, c) as the projection of P onto the XZ plane. Similarly, if we draw a perpendicular line from P towards the YZ plane and the XY plane, we get R(0, b, c) and S(a, b, 0) are the projections of P onto the YZ plane and the XZ plane.

## Equation

For each coordinate plane, we can write equations to represent the plane. For the XY plane, the equation is z=0, for the XZ plane, the equation is y=0, and for the YZ plane, the equation is x=0.

Therefore, the XY plane can be defined as $\left\{\left(x,y,0\right)|x,y\in \mathbb{R}\right\}$, the YZ plane can be defined as $\left\{\left(0,y,z\right)|y,z\in \mathbb{R}\right\}$, and the XZ plane can be defined as $\left\{\left(x,0,z\right)|x,z\in \mathbb{R}\right\}$. Sometimes students may consider that for that XZ plane, $y=k$ for some constant $k\ne 0$, but for this type, the plane $y=k$ seems to be parallel to the XZ plane. About this type of parallel plane, we have given a brief discussion below.Let us take $k\ne 0$ as some constant then the plane x=k is parallel to the YZ plane, the plane y=k is parallel to the XZ plane and the plane z=k is parallel to the XY plane.

When a plane is parallel to the XZ plane then the y-coordinate do not vary, only the x-coordinate and the z-coordinate vary. Similarly, if a plane is parallel to the XY plane then the z-coordinate does not vary, only the x-coordinate and the y-coordinate vary, and if a plane is parallel to the YZ plane then the x-coordinate does not vary, only the y-coordinate and the z-coordinate vary.

For example, consider the equation y=5, and x, z be any real numbers, then it can be represented as $\left\{\left(x,y,z\right)|y=5,x,z\in \mathbb{R}\right\}$.

Then y=5 is a vertical plane and is parallel to the XZ plane.

## Direction

The positive and negative symbol denotes the real number is positive or negative along the given direction. The coordinates that lie in the first octant are nonzero coordinates.

## Formula

The distance between two points (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) in space is given by the formula:

$d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}+{\left({z}_{2}-{z}_{1}\right)}^{2}}$

For the XZ plane, the distance between any two points (x_{1} , 0 , z_{1}) and (x_{2} , 0 , z_{2}) is:

$d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({z}_{2}-{z}_{1}\right)}^{2}}$

The midpoint of the line segment with endpoints (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) in space is given by the formula:

$M=\left(\frac{{x}_{1}+{x}_{2}}{2},\frac{{y}_{1}+{y}_{2}}{2},\frac{{z}_{1}+{z}_{2}}{2}\right)$

For the XZ plane, the midpoint of the line segment with endpoints (x_{1}, 0, z_{1}) and (x_{2}, 0, z_{2}) is:

$M=\left(\frac{{x}_{1}+{x}_{2}}{2},0,\frac{{z}_{1}+{z}_{2}}{2}\right)$

## Practice Problem

Find the distance between the points (2, 0, 4) and (-5, 0, 1).

We can find the distance between these two points using the formula,

$d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({z}_{2}-{z}_{1}\right)}^{2}}$, since the y-coordinate in both points is 0 and so

(2, 0, 4) and (-5, 0, 1) lie in the XZ plane.

$d=\sqrt{{\left(-5-2\right)}^{2}+{\left(1-4\right)}^{2}}$

$=\sqrt{{\left(-7\right)}^{2}+{\left(-3\right)}^{2}}$

$=\sqrt{49+9}$

$=\sqrt{58}$

Therefore, $d=\sqrt{58}$.

## Context and Applications

The three-dimensional space is used for calculating the location of stars, satellites, planets, solar system, and other moving bodies in space. It is used for creating video games, designing, and in the construction of building and soon.

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for:

- Bachelor of Science in Physics
- Master of Science in Physics
- Bachelor of Engineering in Mechanical

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