# Xam Idea Class 12 Maths Chapter 12 Solutions: Three Dimensional Geometry

Xam Idea Class 12 Maths Solutions for Chapter 12 ‘Three-dimensional Geometry’ will help you understand the concepts of three-dimensional shapes. Topics included in this chapter are an introduction to three-dimensional geometry, direction cosine and direction ratios of a line, the angle between two lines, the shortest distance between two lines, plane, the distance of a point from P, equation of a line in space, and coplanarity of two lines.

Xam Idea Class 12 Maths Solutions book has a variety of different types of questions in it with complete and step-by-step solutions. This book is designed in line with the latest CBSE syllabus. Xam Idea Class 12 Maths Solutions book provides a good learning experience and includes basic concepts of the chapter, important questions from the previous year’s CBSE exams, and higher-order thinking skill problems (HOTS). This chapter has a total of 88 solved problems in it.

Xam Idea Class 12 Maths Solutions is the finest study material available to practise past exam questions. But to score high in your CBSE exams, you also need a correct approach and a strong fundamental base. Instasolv’s Xam Idea solutions are designed by keeping this in mind. Our solutions will prepare you for the exams by providing the well-structured, easy to follow, and detailed answers to the problems asked in Xam Idea Class 12 Maths ThreeDimensional Geometry.

## Important topics for Xam Idea Class 12 Maths Solutions Chapter 12: Three-dimensional Geometry

Few important topics that are covered in this chapter are below-mentioned:

**Direction cosine of a line: **Let us assume that the directed line OP makes angles α, β, and g with positive x, y, and z-axis respectively, then cos α, cos β, and cos g, are called direction cosines of a line. They are represented by l, m, and n respectively. Therefore, we can say that the l = cos α, m = cos β, and n = cos g. Also, the sum of squares of direction cosines of a line is always 1.

That means, l^{2} + m^{2} + n^{2} = cos^{2} α + cos^{2} β, + cos^{2} g = 1

**Direction ratio of a line: **Numbers proportional to the direction cosines of a line is called as direction ratios of a line. 3 points to be noted are:

If a, b, and c are direction ratio of a line, then

If a, b, and c are direction ratio of a line, then its direction cosines are

Direction ratios of a line PQ passing through the points P(x1, y1, z1) and Q(x2, y2, z2) are x2 – x1, y2 – y1, and z2 – z1 and direction cosines are.

**The angle between two lines: **Vector form and Cartesian form of the angle between two lines are:

Vector form: Angle between the lines

Then,

Cartesian form: If ϴ is the angle between the lines

Then,

Also, angle (ϴ) between two lines with direction cosines, l_{1}, m_{1}, n_{1} and l_{2}, m_{2}, n_{2} is given by cos ϴ = l_{1} l_{2} + m1 m_{2} + n_{1} n_{2}

Or

**The shortest distance between two lines: **Two non-intersecting lines or non-parallel lines are called skew lines and the shortest distance between them will be the perpendicular line between them. The vector form and Cartesian form formula for the shortest distance between two lines are:

Cartesian form

Then, the shortest distance is,

**Coplanar: **If two lines are parallel to each other, then they are called coplanar.

**Plane:** A surface such that a line segment joining any two points of it lies wholly on it, then it is called a plane. A straight line perpendicular to every line lying on a plane is called normal to the plane.

### Exercise Discussion for Xam Idea Class 12 Maths Solutions Chapter 12: Three-dimensional Geometry

Chapter 12 Three-dimensional Geometry, has a total of 88 solved problems in it. The problems are categorized into 4 parts. Each category is further divided into two parts: PYQ (past years questions) and OIQ (objective inventory questionnaires).

**Very Short Questions **

- This section consists of a total of 20 problems out of which 14 problems fall under the PYQ section and the rest 6 problems fall under the OIQ section.
- Very short QA section covers the very simple question, quick and easy to solve questions that can be solved just by applying formulas. Questions like find the direction cosines, equation of planes, and the sum of intercepts fall under this part.

**Short Questions**

- It consists of a total of 8 problems from which 3 problems belong to the PYQ category and 5 belongs to the OIQ category.
- The complexity of problems asked in this section is more compared to a very short QA.
- Questions like find the distance of the point, prove points are collinear, write vector equations comes under this section.

**Long Questions Part I & II**

- Part I & II both consist of 30 problems each which are further divided into PYQ and OIQ. It covers the most complex and twisted problems of the chapter that can’t be solved by just applying a formula.
- Questions covered under this section are finding the coordinates of points, vector equation for passing line, the angle, finding Cartesian as well as vector equation of the planes, and to prove.

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