Class 11 Maths Introduction To Three Dimensional Geometry is a very scoring topic in the latest CBSE syllabus. This chapter introduces the fundamental concept of 3D geometry. Students learn about coordinate axes and coordinate planes through various numericals. They learn how to locate and identify the coordinates of the point in a 3-dimensional rectangular coordinate system. They also calculate the distance between two points using the distance formula, which is further applied in solving problems related to three-dimensional structures. Mid-points of line segments and centroid of a triangle can also be determined using section formula. Here, in Class 11 Maths Introduction To Three Dimensional Geometry NCERT Solutions, you will learn and practice the different concepts of 3D Geometry.

The coordinate plane divides plane space into three dimensions. The three mutually perpendicular axes of the coordinate plane are the x-, y- and z-axis. The coordinate planes are referred to as the XY, YZ, and XZ planes. There are three types of coordinate systems, namely spherical, cylindrical and rectangular coordinate systems.

The position of a point in space is given by its coordinates. The coordinates of the point are measured with respect to the x-, y- and z-axes and are given by a 3-tuple (x, y, z). In order to determine the coordinates of any point, it is essential to choose a fixed three-dimensional coordinate system. Thereafter, perpendiculars are drawn from the point on the three axes. The lengths of the perpendicular on the x-axis, perpendicular on the y axis, and perpendicular on the z-axis are equal to the x-, y- and z-coordinates, respectively.

The distance between any two points in 3D geometry can be calculated using the distance formula.

Let us assume that there are two points, A and B, such that the coordinates of A and B are given by (x₁, y₁, z₁) and (x₂, y₂, z₂), respectively. Then the distance (d) between these two points is given as 

When a line segment is divided in a ratio by a point, the section formula is used to find the coordinates of that point. 

Let us assume that a line segment AB is formed by joining points (x₁, y_1, z₁) and (x₂, y₂, z₂). If a point C divides this line segment in the ratio m:n either internally or externally, section formula can be used to find the coordinates of point C. Section formula is of two types:

• • Section Formula for Internal Division: AB will be divided in the ratio m:n internally if point C lies on it. According to the section formula for internal division, the coordinates of point C can be given as:

• • Section Formula for External Division: AB will be divided in the ratio m:n externally if point C does not lie on it. According to the section formula for external division, the coordinates of point C can be given as:

   

Some key terms associated with 3D Geometry are as follows:

• Origin: The point of intersection of the three coordinate axes is called the origin. The coordinates of the Origin are (0,0,0).

• Octant: The coordinate planes divide the three-dimensional space into eight parts. Each part is called an octant. The octant in three dimensions is an equivalent of a quadrant in two dimensions.  

• Collinear Points: Points that lie on the same line segment are called collinear points.  The collinearity of points can be easily checked with the help of the section formula.

Problem 1: What are the x- and y-coordinates of a point if it is on the z-axis?

Answer: 0, 0

Explanation: Since the point is somewhere on the z-axis, its x- and y-coordinates of this point are 0, 0.

 

Problem 2: The x-coordinate of a point is zero. Find out the plane that contains this point.

Answer: The point is in the YZ plane. 

Explanation: If the point is in the YZ plane, it will have non-zero y- and z-coordinates, but its x-coordinate will be zero.

 

Problem 3: A point lies in the third octant, what are its possible coordinates ( -3, -2, 6) and (-3, -5, -7)?

Answer: ( -3, -2, 6)

Explanation: ( -3, -2, 6) lies in the third octant as its x- and y-coordinates are negative, and z-coordinate is positive. All three coordinates of a point are negative if it lies in the eighth octant.

Problem 4: Find the distance between the points (1, 2, 3) and ( 2, 5, 3). 

Answer: 10

Explanation: Here, (x1, y1, z1) and (x2, y2, z2) are (1, 2, 3) and ( 2, 5, 3), respectively. 

According to the distance formula, the distance between two points is given by:

Problem 5: A line segment is formed by joining the points (-1, 3, 4) and (2, 4, -6). Find the coordinates of the point, which divides it in the ratio 1: 2 internally.

Answer: (0, 4, 1)

Explanation: Here, m: n= 1: 2; and (x1, y1, z1) and (x2, y2, z2) are equal to (-1, 4, 4) and (2, 4, -5) respectively. 

According to the section formula, the coordinates of the point which divides the given line segment internally in the ratio m: n is given by:

 

Discuss the topics covered in Chapter 12 of NCERT Solutions for Class 11 Maths.

Chapter 12 of NCERT Solutions for Class 11 Maths has been designed as per the latest update in the CBSE syllabus. The topics covered are coordinate axes, coordinate planes in three-dimensional space, coordinates of a point in space, distance between two points, and section formula. Clear conceptual understanding and thorough practice of numericals of these topics are crucial to getting good marks in the CBSE exams.

What are the topics of Class 11 Maths Chapter 12- Introduction to Three Dimensional Geometry?

The Class 11 Maths Chapter 12 Introduction to Three Dimensional Geometry for the new academic session 2021-22 are as follows: 

1. Introducing basic ideas of three-dimensional space.

2. Solving problems related to coordinate axes and coordinate planes in three-dimensional space.

3. Determining coordinates of any point in three-dimensional space.

4. Using the distance formula to find the distance between two points. 

5. Using section formula to find coordinates of a point that divides a line segment in a given ratio.

What are the major learnings from these topics?

The real world deals with three-dimensional structures. Since everything around us is three-dimensional, a sound understanding of this chapter is important to evaluate and forecast real-time events. The concepts of coordinate axes and coordinate planes in Three Dimensional Space, Coordinates of a Point in Space, Distance between Two Points, and Section Formula, find applications in advanced Mathematics and sciences such as Physics.  

 

Where can I get the NCERT Solutions for Class 11 Maths Chapter 12?

MSVGo offers precise and accurate 3D Geometry Class 11 Ncert Solutions. Mathematics is a highly scoring subject. Our experts have painstakingly curated step-by-step solutions with in-depth explanations wherever required for 3D Geometry. These solutions cover all types of questions usually asked in CBSE Term-II exams. When practiced regularly, students witness a marked improvement in the time spent to solve each 3D Geometry problem.