{"id":138623,"date":"2022-04-11T12:13:46","date_gmt":"2022-04-11T06:43:46","guid":{"rendered":"https:\/\/www.aakash.ac.in\/blog\/?p=138623"},"modified":"2023-04-03T12:59:22","modified_gmt":"2023-04-03T07:29:22","slug":"3d-geometry-and-vectors-revision-notes-for-cbse-12th-maths-term-2-exam","status":"publish","type":"post","link":"https:\/\/www.aakash.ac.in\/blog\/3d-geometry-and-vectors-revision-notes-for-cbse-12th-maths-term-2-exam\/","title":{"rendered":"3D Geometry and Vectors: Revision Notes For CBSE 12th Maths Term 2 Exam"},"content":{"rendered":"

3D geometry and vectors were introduced in Maths to help students grasp different types of shapes and figures. Almost all items in the real world have three dimensions. Many household items, for example, contain 3D geometry, such as pens, laptops, windows, culinary utensils, and so on. Students have studied the fundamentals of three-dimensional geometry in class 11. In class 12, the advanced version of 3-D geometry is taught. Vector algebra will now be applied to three-dimensional geometry.\u00a0<\/span><\/p>\n

The goal of this 3-dimensional geometry approach is to create a simple yet elegant study structure. <\/span>Important concepts <\/span><\/a>covered in three-dimensional geometry for Class 12 include direction cosine and direction ratios of a line connecting two points. The notes are based on the term 2 CBSE and NCERT syllabuses.\u00a0<\/span><\/p>\n

Students can review Maths Chapter 11 (Three-dimensional geometry) with notes created according to the most recent exam pattern. The students will also study the equations of lines and planes in space under various conditions, the angle between a line and a plane, and the angle between two lines, among other things. To better understand the material, you should practise the problems based on the <\/span>NCERT solutions for class 12 Maths<\/span><\/a>. This chapter has great importance in exams like JEE 2022, JEE Advanced 2022, National Engineering Olympiad, and other <\/span>olympiads<\/span><\/a> in India.<\/span><\/p>\n

Important Notes<\/h3>\n

A student preparing for CBSE 12th Maths term two examinations ought to study the notes mentioned below in this article. These notes are created by subject experts and toppers who are well versed with the examination pattern. Since this term’s paper is subjective, there are chances that at least two questions out of four long answer-type questions will come from this chapter. Hence it is important to go through each note mentioned in the article.<\/span><\/p>\n

Direction Cosine and Direction Ratio<\/h3>\n

\u00a0The direction cosine and direction ratios are the first things taught in CBSE 12th 3D geometry. These can be understood by making a line pass through the origin (0,0). Let angles formed by the line and axes be \u03b1, \u03b2, \u03b3 with x, y, and z-axes. Therefore finding the cosine of these angles will give us the direction cosine of the line. For now, let us consider the<\/span><\/p>\n

x = ka\u00a0<\/span><\/p>\n

y = kb<\/span><\/p>\n

z = kc<\/span><\/p>\n

x<\/span>a<\/span> = <\/span>y<\/span>b<\/span> = <\/span>z<\/span>c<\/span> = k<\/span><\/p>\n

Therefore the direction cosines will be formulated as:<\/span><\/p>\n

m = \u00b1 a<\/span>2<\/span>\/ \u221a(a<\/span>2<\/span> + b<\/span>2<\/span> + c<\/span>2<\/span>)<\/span><\/p>\n

m = \u00b1 b<\/span>2<\/span>\/ \u221a(a<\/span>2<\/span> + b<\/span>2<\/span> + c<\/span>2<\/span>)<\/span><\/p>\n

m = \u00b1 c<\/span>2<\/span>\/ \u221a(a<\/span>2<\/span> + b<\/span>2<\/span> + c<\/span>2<\/span>)<\/span><\/p>\n

Note that two parallel lines have the same set direction cosines if we are provided with a line that doesn\u2019t pass through the origin. The students need to create a new line from the origin parallel to the given line. Now you can find the direction cosines of the new line, which will be the same as the original line.\u00a0<\/span><\/p>\n

\u00a0The relation between the direction cosines is that the sum of squares of the direction cosines must be equal to one. For the line having direction cosines x, y, and z, the relation is\u00a0\u00a0<\/span><\/p>\n

x<\/span>2<\/span> + y<\/span>2<\/span> + z<\/span>2<\/span> = 1<\/span><\/p>\n

Direction Cosine For a Line Between Two Points<\/h3>\n

Suppose there is a line RS such that R(a<\/span>1<\/span>, b<\/span>1<\/span>, c<\/span>1<\/span>) and S(a<\/span>2<\/span>, b<\/span>2<\/span>, c<\/span>2<\/span>). The direction cosine between these points is formulated as:<\/span><\/p>\n

x = a<\/span>2<\/span> – a<\/span>1<\/span>\/ RS\u00a0<\/span><\/p>\n

y = b<\/span>2<\/span> – b<\/span>1<\/span>\/ RS\u00a0<\/span><\/p>\n

z = c<\/span>2<\/span> – c<\/span>1<\/span>\/ RS\u00a0<\/span><\/p>\n

Here RS = \u221a (a<\/span>2<\/span> – a<\/span>1<\/span>)<\/span>2<\/span> + (b<\/span>2<\/span> – b<\/span>1<\/span>)<\/span>2<\/span> + (c<\/span>2<\/span> – c<\/span>1<\/span>)<\/span>2<\/span><\/p>\n

Line Equations\u00a0<\/b><\/p>\n

Equation Of a Line in 3D Space<\/b>\u00a0<\/b><\/p>\n

The equation of a line can be determined in two ways<\/span><\/p>\n