Three Dimensional Geometry

Direction cosine are 

$\begin{array}{l}\frac{-18}{},\frac{+12}{},\frac{-4}{}\\ =\frac{-18}{22},\frac{12}{22},\frac{-4}{22}\\ =\frac{-9}{11},\frac{6}{11},\frac{-2}{11}\end{array}$

Let the angles be α, β, r which are equal  

Let the direction cosines of the line be l, m, n.

$\begin{array}{l}l=cos\alpha ,m=cos\beta ,n=cosr\\ \alpha =\beta =r\\ l^2+m^2+n^2=1\\ co{s}^2\alpha +co{s}^2\beta +co{s}^{2}r=1\\ co{s}^2\alpha +co{s}^2\alpha +co{s}^2\alpha =1\\ 3co{s}^2\alpha =1\\ co{s}^2\alpha =\frac{1}{3}\\ cos\alpha =\frac{1}{}\end{array}$

Let the direction cosine of the line be l, m, n.

Then,

$\begin{array}{l}l=90^{?}=cos90^{?}=0\\ m=cos135=\frac{-1}{}\\ n=cos45^{?}=\frac{1}{}\end{array}$

Students can use these steps for downloading the PDF for 3D Geometry NCERT Solutions given below;

• Visit the Official Page: Chapter 11 Three dimensional Geometry Solutions
• Access the Solutions and Scroll down to download PDF
• Click on "Download Free PDF"
• Save the PDF in their device

NCERT Solutions for 3D Geometry serves as a mathematical foundation developer for all the students. All competition exam aspirant must use NCERT Textbooks and its solutions for better understanding for fundamental concepts in 3D Geometry. Having a proper understanding of foundation knowledge of chapter is important for competitive exams like JEE and NEET. 

There are various reasons to use Shiksha's NCERT Solutions for 3D Geometry. Students can use 3D Geometry class 12 maths Solutions for various purposes like practice, revision, and making notes. These solutions follow the latest CBSE 2025 syllabus and NCERT textbook pattern. Shiksha Solutions for 3D Geometry are accurate and provide step-by-step explanations of all the questions.

Students must know that question papers are prepared on the basis of NCERT Textbook and its questions. Students can practice NCERT Maths Textbooks for most repeated questions of 3D Geometry. Mostly questions are asked based on direction cosines, finding angles between lines, the shortest distance between skew lines, and equations of planes are frequently repeated.

3D Geometry holds good weightage in the class 12 board exams, So students must be aware of the important topics that can be asked in the exams. Class 12  Three Dimensional Geometry chapter covers many important topics like direction cosines and ratios, equations of a line, angles between two lines, skew lines, the shortest distance, and equations of planes. Students can use our NCERT Solutions for Class 12 3D Geometry for clear understanding of these topics.