Three Dimensional Geometry

You must have learn about 2-D geometry or two-dimensional coordinate system. Here you will learn about three dimensional geometry and other related topics such as definition, formulas, problems based on it.

Introduction to Three Dimensional Geometry

When we talk about 2 dimension geometry, it consists of two coordinates say, x and y in the plane. But when we speak about 3 dimensional geometry, it consists of three coordinates say, x, y and z.

Three-dimensional geometry deal with shapes of square, rectangle, cuboid, rhombus, trapezium, parallelogram etc, which has coordinates in three-dimensional space. These shapes or figures have dimensions-length, width and height. You can take examples of 3D geometry from real worlds also.Like, a room of a house has length, width and height as its dimensions.

Three Dimensional Geometry Formulas

Distance Formula

Let P and Q are two points in space which has coordinates ($x_{1},y_{1},z_{1}$) and ($x_{2},y_{2},z_{2}$), respectively.The distance between two points P and Q is defined by then,

$PQ=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}$

Section Formula

1. If a point R divides the line segment joining the two points P($x_{1}, y_{1}, z_{1}$) and Q($x_{2}, y_{2}, z_{2}$) in the ratio of m:n, internally, then, coordinates of R are;
$R( \frac{mx_{2}+ nx_{1}}{m+n}, \frac{my_{2}+ ny_{1}}{m+n}, \frac{mz_{2}+ nz_{1}}{m+n})$
1. If a point R divides the line segment joining the two points P($x_{1},y_{1},z_{1}$) and Q($x_{2},y_{2},z_{2}$) in the ratio of m:n, externally, then, coordinates of R are;
$R( \frac{mx_{2}- nx_{1}}{m-n}, \frac{my_{2}- ny_{1}}{m-n}, \frac{mz_{2}- nz_{1}}{m-n})$

Mid-Point Formula

If R is the mid-point of the segment joining P($x_{1},y_{1},z_{1}$) and Q($x_{2},y_{2},z_{2}$), then m=n=1 and the coordinate R is given by;

$R( \frac{x_{1}+ x_{2}}{2}, \frac{y_{1}+ y_{2}}{2}, \frac{z_{1}+ z_{2}}{2})$

For Centroid of Triangle

Centroid of the triangle whose vertices are ($x_{1},y_{1},z_{1}$), ($x_{2},y_{2},z_{2}$) and ($x_{3},y_{3},z_{3}$), the coordinate R is given by;

$R( \frac{x_{1}+ x_{2}+ x_{3}}{3}, \frac{y_{1}+ y_{2}+ y_{3}}{3}, \frac{z_{1}+ z_{2}+ z_{3}}{3})$

These formulas are the key concepts and are also covered under three dimensional geometry Class 11 syllabus.

Three Dimensional Geometry Problem and Solution:

To get a clear understanding of the three dimensional formulas, you should solve three dimensional geometry problems. Let us give an example problem here.

Problem: Find the distance between the points P(-2,4,1) and Q(1,2,-5).

Solution: Here, $x_{1}$=-2, $y_{1}$=4, $z_{1}$=1, $x_{2}$=1, $y_{2}$=2 and $z_{2}$=-5.

By the distance formula, we know,

PQ=$\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}$

PQ=$\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}$

PQ=$\sqrt{(1-(-2))^{2}+(2-4)^{2}+(-5-1)^{2}}$

PQ=$\sqrt{9+4+36}$

PQ=7 units

Similarly, you can also solve more problems on the above-given formulas by defining the values for each quantity.