Three-Dimensional Geometry Concepts in Mathematics

Explore the concepts of three-dimensional geometry in mathematics, including direction angles, direction cosines, direction ratios, and equations of lines in space. Learn how to find direction cosines and ratios of a line and understand the properties of X, Y, and Z axes. Gain insights into the unique sets of direction cosines and their applications in solving mathematical problems.

• Three-Dimensional Geometry
• Mathematics
• Direction Angles
• Equations of Lines
• Direction Ratios

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1. an International CBSE Finger Print School Coimbatore SUBJECT NAME - 041 MATHEMATICS GRADE-XII UNIT 11 TOPIC THREE DIMENSIONAL GEOMETRY THREE DIMENSIONAL GEOMETRY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY 13/06/23 1

2. THREE DIMENSIONAL GEOMETRY Suppose a line passing through origin is making angles , , with x-axis, y-axis, z-axis respectively then , , are called direction angles, then cosine of these angles cos , cos , cos are called direction cosines of the directed line l. x 13/06/23 2/15 THREE DIMENSIONAL GEOMETRY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

3. If we reverse the direction of l then direction angles are replaced by their supplements' i.e. - , - , - .Then the sign of direction cosines are reversed. In order to have a unique set of direction cosines ,we must take given line as directed line. these unique direction cosines are denoted by l,m,n. If the given line in space does not passes through the origin, then in order to find it s direction cosines we draw a line through the origin and parallel to the given line. 13/06/23 3/15 THREE DIMENSIONAL GEOMETRY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

4. Any three numbers which are proportional to the direction cosines are called direction ratios and are denoted by a,b,c. Note: here , l=ak , m= bk , n=ck therefore if a,b,c are direction ratios of a line then its direction cosines of the line are l b l= m= n= 13/06/23 4/15 THREE DIMENSIONAL GEOMETRY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

5. For any line if a,b,c are direction ratios then ka,kb,kc [k 0] also can be taken as direction ratios. Note: if l,m,n are direction cosines of a line then l +m +n =1 Note: direction cosines of a line passing through two points(x,y,z) and (x ,y ,z ) is given by x -x , y -y , z -z Note: Direction cosines of X axis are 1,0,0 Direction cosines of Y axis are 0,1,0 Direction cosines of Z axis are 0,0,1 13/06/23 5/15 THREE DIMENSIONAL GEOMETRY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

6. Equation of a line in Space 13/06/23 6/15 THREE DIMENSIONAL GEOMETRY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

7. 13/06/23 7/15 THREE DIMENSIONAL GEOMETRY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

8. Angle Between Two lines 13/06/23 8/15 THREE DIMENSIONAL GEOMETRY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

9. Equation of a Plane 9/15 13/06/23 THREE DIMENSIONAL GEOMETRY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

10. 13/06/23 10/15 10 THREE DIMENSIONAL GEOMETRY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

11. 11/15 13/06/23 THREE DIMENSIONAL GEOMETRY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

12. EQUATION OF PLANE THROUGH THE INTERSECTION OF TWO PLANES Consider two planes new plane a x+b y+c z+d =0 1 a x+b y+c z+d =0 2 The equation of plane through the intersection of the two plane is given By: 1+ 2=0 where is a scalar (a x+b y+c z+d )+ (a x+b y+c z+d )=0 13/06/23 12/15 12 THREE DIMENSIONAL GEOMETRY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

13. FINDING EQUATION OF THE PLANE PASSING THROUGH ONE POINT AND SATISFYING TWO MORE CONDITIONS Note:equation of any plane passing through (x ,y ,z ) can be taken as A(x-x )+B(y-y )+C(z-z )=0 where A,B,C are direction ratios of normal to the plane. 13/06/23 13/15 THREE DIMENSIONAL GEOMETRY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

14. Questions: 1. Find the direction cosines of the line passing through the two points ( 2, 4, 5) and (1, 2, 3). 2. Find the vector and the Cartesian equations of the line through the point (5, 2, 4) and which is parallel to the vector 3. Find the distance between the lines l1and l2given by 13/06/23 14/15 14 THREE DIMENSIONAL GEOMETRY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY

15. References https://www.slideshare.net/TarunSingh389/three-dimensional- geometryclass-12 https://www.aees.gov.in/htmldocs/downloads/e-content_06_04_20/3D- PPt%20MODULE%201.pptx https://www.selfstudys.com/books/ncert-new- book/english/12th/mathematics-part-ii/5-three-dimensional- geometry/143840 https://www.studiestoday.com/download-book/ncert-class-12-maths- three-dimensional-geometry-175520.html 13/06/23 15/15 THREE DIMENSIONAL GEOMETRY/041 MATHEMATHICS/MADHANKUMAR A /MATHS/SNS ACADEMY