Chapter 11
Three Dimensional Geometry
Every example and exercise of the chapter, worked in full. Each page gives the question, the complete solution, the reason behind every step, and the basics it rests on.
New to this chapter? Start with the quick lesson: the formulas, the question types and how to crack each one →Examples
- Example 1Direction cosines of a line at , , to the axesworked
- Example 2Direction cosines from the direction ratios worked
- Example 3Direction cosines of the line through and worked
- Example 4Direction cosines of the -, - and -axesworked
- Example 5, , are collinearworked
- Example 6Vector and Cartesian equations of the line through parallel to worked
- Example 7Angle between lines along and worked
- Example 8Angle between two lines in Cartesian form, direction ratios and worked
- Example 9Shortest distance between two skew lines in vector formworked
- Example 10Distance between two parallel lines along worked
Exercise 11.1
- Exercise 11.1 · Q1Direction cosines of a line at , , to the axesworked
- Exercise 11.1 · Q2Direction cosines of a line equally inclined to the axesworked
- Exercise 11.1 · Q3Direction cosines from the direction ratios worked
- Exercise 11.1 · Q4, , are collinearworked
- Exercise 11.1 · Q5Direction cosines of the sides of the triangle , , worked
Exercise 11.2
- Exercise 11.2 · Q1Three lines with given direction cosines are mutually perpendicularworked
- Exercise 11.2 · Q2Line through and is perpendicular to the line through and worked
- Exercise 11.2 · Q3Line through and is parallel to the line through and worked
- Exercise 11.2 · Q4Line through parallel to worked
- Exercise 11.2 · Q5Vector and Cartesian equations of the line through along worked
- Exercise 11.2 · Q6Cartesian equation of the line through parallel to worked
- Exercise 11.2 · Q7Vector form of worked
- Exercise 11.2 · Q8Angle between two pairs of lines in vector formworked
- Exercise 11.2 · Q9Angle between two pairs of lines in Cartesian formworked
- Exercise 11.2 · Q10Values of that make two lines perpendicularworked
- Exercise 11.2 · Q11 and are perpendicularworked
- Exercise 11.2 · Q12Shortest distance between lines through and worked
- Exercise 11.2 · Q13Shortest distance between two lines in Cartesian formworked
- Exercise 11.2 · Q14Shortest distance between lines through and worked
- Exercise 11.2 · Q15Shortest distance between two lines with parameters and worked
Miscellaneous Exercise
- Miscellaneous · Q1Angle between lines with direction ratios and worked
- Miscellaneous · Q2Equation of the line through the origin parallel to the -axisworked
- Miscellaneous · Q3Value of that makes two lines perpendicularworked
- Miscellaneous · Q4Shortest distance between lines through and worked
- Miscellaneous · Q5Vector equation of the line through perpendicular to two given linesworked
Want an email when new chapters are added?
Leave your email and we will write when a new chapter or a new set of worked solutions is ready. That is all we use it for, and you can ask to be taken off the list at any time.
You are on the list. We will email you when new chapters are added, and for nothing else.