An Overview of Three Dimensional Geometry in Class 12
Three dimensional geometry includes figures with length, breadth and height. Unlike two dimensional shapes, these shapes have depth or thickness. Moreover, these shapes consists of three coordinates, x-axis, y-axis, and z-axis. Additionally, in three-dimensional geometry, you need these three parameters to find the exact location of a point.
Furthermore, in mathematics, usually, we come across Cartesian geometry. It primarily deals with two-dimensional objects, and have two coordinates x and y-axis. On the other hand, three dimensional geometry has three coordinates to operate with.
Chapter 11 - Three - Dimensional Geometry
In this chapter, you will study the direction of cosines and a line joining two points. Moreover, you will also learn about the equation of lines and planes in space, under varied conditions. Additionally, the angle between two lines, planes as well as a line and a plane. You will also study the minimum distance between two skewed lines and the distance of one point from a plane.
Furthermore, the results of the above-mentioned topics are mostly in vector forms. Nonetheless, you will also understand the ways to convert these results into the Cartesian form. This form offers more clarity on a geometric picture of a situation.
Topics of Chapter 11
11.1 - Introduction
11.2 - Direction cosines and direction ratios of a line
11.3 - Equation of a line in space
11.4 - Angle between two lines
11.5 - The shortest distance between two lines
11.6 - Plane
11.7 - Coplanarity of two lines
11.8 - Angle between two lines
11.9 - Distance of a point from a p
11.10 - Angle between a line and a plane
Important Areas of Three Dimensional Geometry
11.2 - Direction cosines and direction ratios of a line
As you have already covered in chapter 10, a direct line passing through the origin produce angles alpha, beta and gamma with x-axis, y-axis and z-axis. Moreover, these angles are known as direction angles. Therefore, the cosine of these angles is cos alpha, cos beta, and cos gamma. Hence, these angles are called cosine of the directed line L.
Therefore, if you reverse the direction L, it will be replaced by its supplements. Furthermore, you should know that any line in space can be extended in the opposite direction. Thus, they have two different sets of cosines.
However, to have an exclusive set of cosines for a particular line in space, you should consider the given line as a directed one. Such unique direction cosines are symbolised by I, m and n.
11.3 - Equation of a line in space
In three dimensional geometry, a line is considered unique if –
It passes through a particular point and has a specific direction.
Or it passes through two specified points.
11.5 - The shortest distance between two lines
In case two lines in space overlap at a given point, then the shortest distance between them will be zero. However, if the two lines are parallel, then the perpendicular distance between them is the shortest one. It means the size of the perpendicular drawn between these two lines represents their distance.
11.6 - Plane
A plane is decided by one of the following aspects –
It goes through a point and is perpendicular to a given distance.
It passes through three non-collinear points.
Equation of a plane in standard form, i.e. normal to the plane and its distance from the source is provided.
Vedantu, Your Partner for Learning
Three dimensional geometry is one of the most important chapters of mathematics in class twelve. In case you are having difficulties navigating through this chapter, you can visit the official website of Vedantu to avail study materials that will help you to understand this chapter. Moreover, you can join the live classes with subject experts to clear your doubts. You access these services via the official app of Vedantu.
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FAQs on Three Dimensional Geometry
1. What is Three Dimensional Geometry?
Ans. Three dimensional geometry deals with 3-D figures instead of 2-D figures. In mathematics, typically, you learn about Cartesian geometry which is primarily two dimensional as it has two axis points x and y.
However, three-dimensional figures have three coordinates to operate with. These are, x-axis, y-axis, and z-axis. Hence, three-dimensional geometrical figures have length, breadth and height. Unlike two dimensional shapes, these shapes have depth or thickness. Furthermore, with the help of its three parameters, you can find the precise location of a point.
2. What are Practical Uses of Three-Dimensional Geometry?
Ans. Three dimensional geometry has a plethora of practical uses. Architecture is probably the biggest example here. This field takes assistance of geometry to define the spatial form of construction. Moreover, it creates construction blueprint with the help of geometry.
Additionally, three-dimensional geometry allows architects to create scale models of their proposed building with proper proportions. Video game development and robotics are also influenced by geometry. The way characters in a video game moves is influenced by geometrical computations. Video game engines use raycasting; this technique allows them to simulate three-dimensional figures using a two-dimensional map.
3. What is Geometry?
Ans. Geometry is a subdivision of mathematics that deals with the shape of various objects. Besides, it is one of the ancient branches of mathematics. The rise of this section of mathematics was mainly to resolve practical problems. The term geometry has derived from Greek words, geo (earth) and metron (measure). Additionally, this section primarily deals with 2-D and 3-D figures. However, the real-life applications of geometry have increased over the years. Nowadays, 2-D and 3-D shapes are being used to develop various abstract ideas.
4. What are the Subfields of Contemporary Geometry?
Ans. There are several subfields of contemporary geometry. At first, the Euclidean geometry comprises the study of points, planes, line, angles, triangles, circles, etc. It has applications in computer science, crystallography and different branches of contemporary mathematics. Moreover, differential geometry uses linear algebra and calculus to resolve problems of geometry.
Additionally, topology studies the property of geometric objects, which remain unchanged after regular mapping. It is mainly used to study the properties of space. Furthermore, convex geometry examines convex shapes via real analysis technique. It is primarily used in number theory. Algebraic geometry studies this subject with the help of algebra. It is mostly used in cryptography.
