NCERT Books Free Download for Class 12 Maths Chapter-11 Three Dimensional Geometry

Students who study Chapter 11- Three Dimensional Geometry Utilising the NCERT Books Will be Able to Comprehend the Following:

• A line connecting two places has direction cosines and direction ratios. 

• The shortest distance between two lines, coplanar and skew lines, and the Cartesian equation and vector equation of a line. 

• A plane's Cartesian and vector equations. 

• The angle is formed by two lines, two planes, or a line and a plane. 

• A point's distance from a plane.

Important Questions From Previous Years

• Find the Cartesian and Vector equations for the line that goes through the point (-2, 4, -5) and is parallel to (x+3)/3 = (y-4)/5 = (8-z)/-6. (2017 CBSE Sample Paper)

• Find the vector equation of the line parallel to the line 5x – 25 = 14 – 7y = 35z and pass-through point A(1, 2, –1). (Delhi, India, 2017)

• The centroid of triangle ABC is the point (α, β, γ) where a plane intersects the coordinate axes in A, B, and C. Show that the plane's equation is x/α + y/β + z/γ = 3. (2017 CBSE Sample Paper)

• If the vectors p = ai + j + k, q = I + bj + k, and r = I + j + ck are all coplanar, then 1/(1-a) + 1/(1-b) + 1/(1-c) = 1 for a, b, and c ≠ 1. (2017 CBSE Sample Paper)

Topics Covered By the Chapter Three Dimensional Geometry

According to the current CBSE Syllabus 2024-25, the chapter Three Dimensional Geometry accounts for 14 of the total marks. This chapter includes three activities, as well as a random, practice to assist students to grasp the fundamentals of Three Dimensional Geometry. The following are some of the topics covered in Chapter 11 of NCERT Books for Class 12 Maths:

• The cosines of the angles formed by a line with the positive directions of the coordinate axes are called the direction cosines of a line.

• If a line's direction cosines are l, m, and n, then l² + m² + n² = 1.

• A line's direction ratios are the numbers that are proportional to the line's direction cosines.

• Skewed lines are lines that are neither parallel nor intersecting in space. They are positioned on distinct planes.

• The angle between skew lines is the angle formed by two intersecting lines drawn parallel to one another from any point (ideally through the origin).

• If l₁, m₁, n₁, and l₂, m₂, n₂ are the direction cosines of two lines, and is the acute angle between them, cos = |l₁l₂ + m₁m₂ + n₁n₂|