Three Dimensional Geometry

Three Dimensional Geometry uses vectors to describe lines in space, combining the coordinate geometry of Class 11 with the vector algebra of the previous chapter. In this chapter you will learn the direction cosines l, m, n of a line, the angles it makes with the coordinate axes, and the result l^2 + m^2 + n^2 = 1. You will also use direction ratios, which are numbers proportional to the direction cosines, and find the direction cosines of the line through two given points. You will write the equation of a line in vector form, r = a + lambda b, and in Cartesian form, through a given point and parallel to a given vector, or through two given points. You will find the angle between two lines from their direction ratios and state the conditions for lines to be parallel or perpendicular. The chapter ends with skew lines, which neither intersect nor are parallel, and the formulas for the shortest distance between two skew lines and between two parallel lines.

Detailed Worksheet: Three Dimensional Geometry Section A - Definitions (10 marks) 1. Define the direction cosines of a line. Prove that l^2 + m^2 + n^2 = 1. (2 marks) 2. What are direction ratios? How are direction cosines obtained from direction ratios? (2 marks) 3. Write the vector and Cartesian equations of a line passing through a point with position vector a and parallel to a vector b. (2 marks) 4. What are skew lines? Give one example from a room. (2 marks) 5. State the conditions for two lines with direction ratios a1, b1, c1 and a2, b2, c2 to be (i) perpendicular and (ii) parallel. (2 marks) Section B - Calculations and Applications (15 marks) 6. Find the direction cosines of a line with direction ratios 2, -1, -2. (3 marks) 7. Find the direction cosines of the line passing through the points (-2, 4, -5) and (1, 2, 3). (3 marks) 8. Find the vector and Cartesian equations of the line through the point (5, 2, -4) and parallel to the vector 3i + 2j - 8k. (3 marks) 9. Find the angle between the lines (x + 3)/3 = (y - 1)/5 = (z + 3)/4 and (x + 1)/1 = (y - 4)/1 = (z - 5)/2. (3 marks) 10. Show that the points P(2, 3, -4), Q(1, -2, 3) and R(3, 8, -11) are collinear. (3 marks) Section C - Diagrams (10 marks) 11. Draw a line OP in space making angles alpha, beta and gamma with the x, y and z axes. Mark the direction cosines and show that l = x/r, m = y/r and n = z/r for P(x, y, z). (4 marks) 12. Draw a diagram of two skew lines and show their shortest distance as the length of the common perpendicular. (3 marks) 13. Draw a diagram showing the line r = a + lambda b, marking the fixed point A with position vector a, the direction vector b and a general point R. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Find the shortest distance between the lines r = i + 2j + k + lambda(i - j + k) and r = 2i - j - k + mu(2i + j + 2k). (5 marks) 15. Find the distance between the parallel lines r = i + 2j - 4k + lambda(2i + 3j + 6k) and r = 3i + 3j - 5k + mu(2i + 3j + 6k). (5 marks) 16. Find the value of p so that the lines (1 - x)/3 = (7y - 14)/(2p) = (z - 3)/2 and (7 - 7x)/(3p) = (y - 5)/1 = (6 - z)/5 are at right angles. Explain how you rewrote each line in standard form. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Write vectors using i, j, k and show the standard form of each line before using formulas.