In solid geometry, an octant is one of the eight divisions of a three-dimensional Euclidean coordinate system that are determined by the signs of the coordinates. The two-dimensional quadrant and the one-dimensional ray are analogous to it. The space is divided into eight sections known as octants by the three mutually perpendicular coordinate planes. The sign of a point’s coordinates depends on the octant in which it is located. All of the coordinates are positive in the first octant and negative in the seventh.

In this math article, we will learn about Octant definition, sign conventions, representation and solved examples in detail.

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What is an Octant?

The space is divided into eight sections by the three mutually perpendicular coordinate planes, with each section being referred to as an octant.

One of the eight spaces that can have any of the (+/-, (+/-, (+/-) sign combinations for x, y, and z.

An octant is typically named by listing all of its signs, such as (+,+,), or (+,+,). Although identical ordinal name descriptors are not provided for the other seven octants, octant (+,+,+) is occasionally referred to as the first octant.

How to Plot the Points in 3D using octants?

A plane divided into four quadrants is referred to as having a rectangular coordinate system, as seen in the illustration on the left.

An origin and six open axes make up the Cartesian coordinate system, which is used to represent three-dimensional space; +z and -z are perpendicular to the x-y plane.

These axes define three planes that, as shown in the illustration to the right, divide the space into eight sections known as octants.

Three directions are shut off by these planes: front to rear, top to bottom, and left to right.

According to tradition, the x-y plane’s four quadrants are numbered as follows:

Quadrant 1 points have coordinates of +x and +y, quadrant 2 points of -x and +y, quadrant 3 points of -x and -y, and quadrant 4 points of +x and -y.

The octants in three-dimensional space do not yet have a defined numbering system, although most people consider the area bounded by +x, +y, and +z to be the first octant.

What is the first octant ?

The first octant is the one where each of the three locations is positive.

The first octant is the area beneath the xyz axis where the values of all three variables are positive. The first octant is one of the eight divisions established by the coordinate signs in a three-dimensional Euclidean coordinate system.

For example, the first octant has the points (2,3,5).

Representation of Vectors and Lines in the First Octant?

We only need to add one more number to a line to represent a vector, but we need to understand direction ratios and direction angles to express a vector as a line.

Consequently, let’s first represent an arbitrary position vector, denoted by the notation \(\vec{a} = \left ( a_{x}, a_{y}, a_{z}\right )\), in the first octant, as seen below;

Finding the Position of a Point in the First Octant

The distance OP is 12 units, and the position vector \(\vec{OP}\) forms angles \( \alpha = 45^{\circ}\) and \( \beta = 60^{\circ} \) with the x and y axes, respectively, when one is requested to find the point \(P\left ( x,y,z \right )\) as shown in the above figure.

Use the following formula to determine where the point \(P\left ( x,y,z \right )\) is located.

\(l^{2} + m^{2} + n^{2} = \cos^{2}\alpha + \cos^{2}\beta + \cos^{2}\gamma = 1\)

Given that \(\alpha \) and \(\beta \) are known, we can now determine \(\gamma\):

\(\cos^{2}45^{\circ} + \cos^{2}60^{\circ} +\cos^{2}\gamma= 1\)

\(\left ( \frac{1}{\sqrt{2}} \right )^{2} + \left ( \frac{1}{2} \right )^{2} = 1-\cos ^{2}\gamma\)

\(\sin ^{2}\gamma = \frac{3}{4}\)

\(\sin ^{2}\gamma = \frac{\sqrt{3}}{2}\)

\(= \sin 60^{\circ}\)

As a result, when comparing the sine argument, we obtain

\(-\gamma = 60^{\circ}\)

\(P\left ( x,y,z \right ) = \left ( OP \cos \alpha, OP\cos \beta , OP\cos \gamma \right )\)

\(\left ( 12.\frac{1}{\sqrt{2}},12.\frac{1}{2},12.\frac{1}{2}\right )\)= \( \left ( 6\sqrt{2},6,6 \right )\)

Calculus in the First Octant

Finding the volumes of three-dimensional objects with a given surface and a set of coordinate planes is simple.

For example: By using the coordinate planes and the cylinder \( x^{2} + y^{2} = 9\) and x + z = 9, one may determine the volume of the solid that is formed:

First, consider the following illustration based on the scenario provided:

A triple integral serves as the volume’s representation. The bounded region’s volume should be expressed as –

\(v = \iint_{R} d V\)

R is the region that is specified and is given by \(R = \left \{ \left ( x,y,z \right ) x,y,z >0;x^{2}+y^{2}\leq 9;z<9-x\right \}\)

The volume is obtained by integrating the surface integral. So let’s move forward as – As a surface integral, transform the above integral.

\(v = \int_{a}^{b}\int_{c}^{d} f\left ( z \right )dx dy = \int_{a}^{b}\int_{c}^{d}\left ( 9-x \right )dxdy\)

Then, observe the reflection of the elliptical surface on the -xy plane, which is a circle with radius zero, to establish the limits of integration. Volume integral then becomes,

\(v = \int_{a}^{b}\int_{c}^{d} \left ( z \right )dx dy = \int_{0}^{3}\int_{0}^{\s \(\) \int_{0}^{3}\int_{0}^{\sqrt{9-y^{2}}} \left ( 9-x \right )dxdy = \int_{0}^{3}\left ( \left [ 9x – \frac{1}{2} x^{2}\right ] ^{\sqrt{9-y^{2}}}_{0}\right ) dy\)

= \(\int_{0}^{3}\left ( \left [ 9\left ( \sqrt{9-y^{2}} \right )-\frac{1}{2}\left ( \sqrt{9-y^{2}} \right ) \right ] \right )dy\)

Consequently, if one integrates it correctly, the volume will be

\(V = \int_{0}^{3} \int_{0}^{\sqrt{9-y^{2}}}\left ( 9-x \right )dxdy = 54.617\)

Properties of Octant in 3D Geometry

 1. Division of 3D Space

• The 3D coordinate system is divided into 8 regions called octants by the x, y, and z axes.
   

• These divisions are formed by the planes:
   

  • XY-plane (z = 0)
     

  • YZ-plane (x = 0)
     

  • XZ-plane (y = 0)

 2. Each Octant Has a Unique Sign Combination

Each octant corresponds to a different combination of signs of (x, y, z):

3. First Octant is Most Commonly Used

• The first octant (x > 0, y > 0, z > 0) is commonly used for plotting points and solving real-life 3D problems.

 4. Points on Axes or Planes Don't Belong to Any Octant

• If any coordinate is zero, the point lies on an axis or plane, not inside an octant.

  • e.g., (3, 0, 5) lies on the XZ-plane, not in any octant.

 5. Symmetry

• Octants are symmetrical with respect to the coordinate planes.

• Opposite octants have opposite signs for all coordinates.

Applications of Octants in 3D Geometry

 1. Identifying Point Positions in 3D Space

• Octants help classify where a point lies in 3D space based on the sign of its coordinates (x, y, z).
• This is fundamental in geometry, physics, and computer graphics.

 2. 3D Graphing and Plotting

• In 3D coordinate systems, plotting points, vectors, or functions accurately requires knowing which octant the point lies in.
• Octants provide a framework for better visualization of 3D data.

 3. Computer Graphics & Animation

• In 3D modeling software and game engines, octants are used to partition space for efficient rendering.
• Helps in collision detection, object placement, and perspective viewing.

 4. Physics and Engineering

• In fields like mechanics, electromagnetism, and structural analysis, octants help in understanding the direction and behavior of forces or vectors in 3D.
• Used in CAD tools for designing 3D structures.

5. Medical Imaging & 3D Scanning

• Octant-based division is used in MRI, CT scans, and 3D reconstruction to divide and analyze parts of the human body or objects.

Summary

• The three mutually perpendicular coordinate planes divide the space into eight parts called octants.
• There are 8 octants.
• One of the eight places where the (+/-, (+/-), and (+/-) sign combinations for x, y, and z can be used.
• Each of the three sites is positive in the first octant.
• The octant in which a location lies can be determined using the signs of the coordinates.

Solved Questions

Problem: 1Identify the octants in which the points (2, – 4, -7) are located.

Solution:

In this case, y is negative, z is negative, and x is positive. It is hence in Octane VIII.

Problem: 2 Which octant does (4, 2, 3) belong in?

Solution:

One of the sites where the eight quadrants of the plane are divided is where the three perpendicular coordinate axes converge.

All the points are positive in the first quadrant, x is negative in the second, and the final two are positive, and so on.

Therefore, the point (4,2,3) is in the second octant.

Problem: 3 The following points are located in which octants? (1, 2, 3),</[>

Solution:

Point (1, 2, 3) has a positive x-coordinate, y-coordinate, and z-coordinate. This point is therefore in octant I.

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