Three Dimensional Geometry Formulas Distance Section Formula and Solved Examples
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.
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FAQs on Three Dimensional Geometry Concepts and Applications in Space
1. What is Three Dimensional Geometry?
Three Dimensional Geometry is the branch of mathematics that studies points, lines, and planes in three-dimensional space (x, y, z). It extends coordinate geometry to space where every point is represented as (x, y, z). In 3D geometry, we analyze distances, angles, direction ratios, direction cosines, and equations of lines and planes. It is widely used in physics, engineering, and computer graphics.
2. What is the distance formula in Three Dimensional Geometry?
The distance between two points in 3D space is given by √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]. If A(x₁, y₁, z₁) and B(x₂, y₂, z₂) are two points, then:
- Distance AB = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]
3. What is the section formula in Three Dimensional Geometry?
The section formula gives the coordinates of a point dividing a line segment in a given ratio in 3D space. If point P divides AB internally in the ratio m:n, then:
- P = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n), (mz₂ + nz₁)/(m+n))
4. What is the equation of a line in Three Dimensional Geometry?
The equation of a line in 3D passing through point (x₁, y₁, z₁) with direction ratios (a, b, c) is written in symmetric form as (x−x₁)/a = (y−y₁)/b = (z−z₁)/c. It can also be written in vector form as:
- r = a + λb
5. What is the equation of a plane in Three Dimensional Geometry?
The general equation of a plane in 3D geometry is Ax + By + Cz + D = 0. Here, A, B, and C are the direction ratios of the normal to the plane. Example: 2x + 3y − z + 5 = 0 represents a plane whose normal vector is (2, 3, −1).
6. How do you find the angle between two lines in 3D?
The angle between two lines in 3D is found using their direction ratios and the formula cosθ = (a₁a₂ + b₁b₂ + c₁c₂) / (√(a₁²+b₁²+c₁²) √(a₂²+b₂²+c₂²)). Steps:
- Identify direction ratios of both lines.
- Apply the dot product formula.
- Find θ using cos⁻¹.
7. What is the distance of a point from a plane?
The distance of a point (x₁, y₁, z₁) from the plane Ax + By + Cz + D = 0 is |Ax₁ + By₁ + Cz₁ + D| / √(A² + B² + C²). This formula gives the shortest (perpendicular) distance from the point to the plane. It is widely used in vector geometry and 3D coordinate problems.
8. What is the difference between direction ratios and direction cosines?
Direction ratios are proportional numbers indicating a line’s direction, while direction cosines are the cosines of angles made with coordinate axes.
- Direction Ratios (a, b, c): Any proportional set representing direction.
- Direction Cosines (l, m, n): cosα, cosβ, cosγ with condition l² + m² + n² = 1.
9. How do you find the shortest distance between two skew lines?
The shortest distance between two skew lines is given by |(a₂−a₁) · (b₁ × b₂)| / |b₁ × b₂|, where b₁ and b₂ are direction vectors. Steps:
- Find direction vectors of both lines.
- Compute their cross product.
- Apply the scalar triple product formula.
10. What are the main applications of Three Dimensional Geometry?
Three Dimensional Geometry is mainly used to model and analyze objects in space and real-world structures. Key applications include:
- Engineering design and architecture
- Computer graphics and 3D animation
- Physics and mechanics
- Navigation and aerospace calculations
