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.
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FAQs on Three Dimensional Geometry
1. What is meant by Three Dimensional Geometry in the context of Class 12 Maths?
Three Dimensional Geometry refers to the study of geometric figures having three dimensions—length, breadth, and height. In this chapter, all positions are described using x, y, and z coordinates, and the logic extends from two-dimensional Cartesian geometry to space. Students learn to analyse lines, planes, and their intersections as per the CBSE 2025–26 syllabus.
2. How do direction cosines and direction ratios define the orientation of a line in three-dimensional space?
In three-dimensional geometry, direction cosines (l, m, n) are the cosines of the angles that a directed line makes with the positive x-, y-, and z-axes. Direction ratios are proportional to the direction cosines and give the ratios of the direction numbers along each axis. These help uniquely define the direction of a line in space.
3. What forms can the equation of a plane take in Three Dimensional Geometry?
The equation of a plane can be expressed in several forms, including:
- Normal form: ax + by + cz = d, where (a, b, c) is the normal vector.
- Intercept form: x/a + y/b + z/c = 1, where a, b, and c are intercepts on the axes.
4. How do you find the shortest distance between two skew lines in space?
To calculate the shortest distance between two skew lines (lines that do not intersect and are not parallel), use the formula:
Distance = |(a2 - a1) • (b1 × b2)| / |b1 × b2|
where a1 and a2 are position vectors of points on each line, and b1 and b2 are their respective direction vectors.
5. Why is the concept of coplanarity important in Three Dimensional Geometry?
The concept of coplanarity helps determine whether given lines or points lie on the same plane. In many problems, confirming coplanarity is essential to simplify calculations and to use properties of plane geometry, especially when analysing intersections and relative positions in space.
6. How can the angle between two lines be calculated in three dimensions?
The angle θ between two lines with direction vectors b₁ and b₂ can be found using:
cos θ = (b₁ • b₂) / (|b₁| |b₂|)
This formula utilises the dot product to measure how much the direction vectors align with each other in space.
7. What are common misconceptions students face while working with equations of lines and planes in Three Dimensional Geometry?
- Confusing direction cosines with direction ratios, leading to incorrect equations of lines.
- Assuming every two lines in space intersect, whereas skew lines never meet.
- Applying two-dimensional geometry logic to three-dimensional problems, which can result in misinterpretation of spatial relations.
8. How do you find the distance of a point from a plane, and why is it significant for board exams?
Given plane equation ax + by + cz + d = 0 and point (x₁, y₁, z₁), the distance is:
Distance = |ax₁ + by₁ + cz₁ + d| / √(a² + b² + c²)
This concept often appears in 3- or 5-mark questions as it tests both understanding of planes and algebraic manipulation, key for CBSE board marks.
9. What is the relevance of vector and Cartesian forms in Three Dimensional Geometry problems?
Vector form provides a concise way to express positions and operations, making proofs and manipulations easier. Cartesian form translates these vectors into coordinate equations, which are more visual and practical for calculations. Both forms are integral to CBSE Class 12 exam questions for Three Dimensional Geometry.
10. How does understanding Three Dimensional Geometry support real-life applications such as engineering or video game design?
Knowledge of Three Dimensional Geometry enables students to model objects in space, crucial for engineering blueprints, architectural design, robotics, and simulation environments in video games. Concepts like spatial coordinates, angles, and distances are foundational in creating accurate digital and physical models.
