Section Formula: 3D Coordinate System

Points are geometry’s most fundamental characteristics, and they lack dimensions. The connection of two points can occur by using exactly one straight line. A point on this straight line undertakes the division of the line into two parts. In two-dimensional geometry, the point’s coordinates undertake division of line segments either internally or externally. The calculation of this internally or externally division can take place in a particular ratio by using the coordinates of the endpoints. The extension of a similar formula can take place into 3D space or a 3D coordinate system. This refers to the section formula. Let us know more about the section formula: 3D coordinate system. One must also refer to the 3D coordinate system graph to better understand.

What is Three-dimensional Geometry

It is crucial to understand three-dimensional before diving into ‘section formula: 3D coordinate system’. 3D geometry or three-dimensional geometry is one of the most critical tools in any scientist or engineer’s toolkit. One can understand a three-dimensional space in terms of 3 coordinates- x, y, and z. It involves a geometric setting in which there is a requirement of three values to determine the element’s position.

The expression of three-dimensional geometry takes place in model real-world quantities like electrical signals, fluid flows, velocity, and several other physical quantities. 

Understanding Section Formula in 3D

Now let us try to understand this formula in a 3D coordinate system. In two dimension geometry, the implementation of the section formula takes place to determine a point’s coordinates. Moreover, this point undertakes the division of a line segment internally in a specific ratio. 

To determine the position of a point in space, a 3D coordinate system is a must. After selecting a fixed 3D coordinate system, any point P’s coordinates in that system can be given.

In the case of a rectangular coordinate system, its arrangement is in the form of an ordered 3-tuple (x, y, z). Furthermore, if you already know the coordinates (x, y, z), it becomes easy to locate the point P in space. The extension of the concept of section formula can take place to 3D geometry to determine the coordinates of a point that undertakes division of a line in a certain ratio.

Remember, for the best understanding of this topic, you require a 3D coordinate system graph. 

Now, consider the following two points:

• A (x1, y1, z1) 
• B(x2, y2, z2)

Consider a point P(x, y, z) that undertakes division of AB in the ratio m:n

For the determination of the coordinates of the point P, follow the steps mentioned below:

• Draw AL, PN, and BM such that it is perpendicular to the XY plane in a way that AL || PN || BM
• The lying of the points L, M and N must take place on the straight line whose formation occurs due to the intersection of a plane involving AL, PN and BM and XY- plane
• From the point P, draw a line segment ST so that it becomes parallel to LM
• Now, the intersection of AL takes place by ST in an external fashion at S, and it undertakes the intersection of BM at T in an internal fashion

Now, ST is such that it is parallel to LM and AL || PN || BM. As such, the qualification of the quadrilaterals LNPS and NMTP take place as parallelograms.

Also, ∆ASP ~∆BTP therefore,

m/n = AP/BP = AS/BT = SL – AL/BM – TM = NP – AL/BM – PN = z – z1/z2 – z

On performing a rearrangement of the above equation we get,

Mz2 + nz1/m + n

The above procedure’s repetition can occur if you draw perpendiculars to XZ and YZ- planes to achieve the point P’s x and y coordinates. The point P undertakes the division of the line segment AB in the ratio m:n in an internal fashion.

• x = mx2 + nx1/m + n
• y = my2 + ny1/m + n

Section Formula in an Internal Fashion

Thus, the coordinates of the point P(x, y, z) undertakes the division of the line segment joining the points A(x1, y1, z1) and B(x2, y2, z2) in the ratio m:n in an internal fashion is given by:

• mx2 + nx1/m + n
• my2 + ny1/m + n
• mz2 + nz1/m + n

Section Formula in an External Fashion

If the given point P undertakes the division of the line segment that joins the points A(x1, y1, z1) and B(x2, y2, z2) in an external fashion in the ratio m:n, then the P’s coordinates of P can be represented by the replacement of n with –n as:

• mx2 – nx1/m + n
• my2 – ny1/m + n
• mz2 – nz1/m + n

This is representative of the section formula for 3d geometry. 

Conclusion

Points are geometry’s most fundamental characteristics, devoid of dimensions. In two-dimensional geometry, the point’s coordinates undertake division of line segments either internally or externally. The calculation of this internally or externally division can take place with the help of a formula. The extension of a similar formula can take place to 3D space and is known as the section formula. It is also essential to understand three-dimensional geometry and its usage when learning about this formula. Therefore, one must also learn this formula in both the internal and external fashion.