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Direction cosines and ratios of a line joining two points sheet
3-mark answer frame

Show that the points A(2, −1, 3), B(4, 3, 1) and C(3, 1, 2) are collinear. Find the direction cosines of the line through them.

How to lay out the answer. Follow the frame, then write it in your own words.

  1. 1 · Write what is given and what is to be found. Given: A(2, −1, 3), B(4, 3, 1), C(3, 1, 2). To show: A, B, C are collinear. To find: the DCs of the line.
  2. 2 · Draw the figure or sketch. Draw a straight line and mark A, C and B on it, with C between A and B.
  3. 3 · Write the formula. DRs of the line through (x₁, y₁, z₁) and (x₂, y₂, z₂) are x₂ − x₁, y₂ − y₁, z₂ − z₁.
  4. 4 · Find and compare the DRs.
    • DRs of AB: 4 − 2, 3 − (−1), 1 − 3 = 2, 4, −2
    • DRs of AC: 3 − 2, 1 − (−1), 2 − 3 = 1, 2, −1
    • 2/1 = 4/2 = −2/−1 = 2, so the DRs are proportional and AB is parallel to AC.
    • A is a common point, so A, B, C are collinear.
  5. 5 · Find the direction cosines. Use DRs 1, 2, −1: √(1 + 4 + 1) = √6. DCs: 1/√6, 2/√6, −1/√6.
  6. 6 · Check and box the answer. C is the midpoint of AB: ((2 + 4)/2, (−1 + 3)/2, (3 + 1)/2) = (3, 1, 2). Collinear; DCs 1/√6, 2/√6, −1/√6.

Scoring tip: Proportional DRs alone show only that the lines are parallel. Write 'A is a common point' to earn the last mark.