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Components of a vector sheet
3-mark answer frame

Show that the points A(2, 1, −1), B(4, 4, 1) and C(8, 10, 5) are collinear. Find |AB⃗| and |BC⃗| and check that |AC⃗| = |AB⃗| + |BC⃗|.

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, −1), B(4, 4, 1), C(8, 10, 5). To show: A, B, C are collinear. To find: |AB⃗|, |BC⃗|, |AC⃗|.
  2. 2 · Write the rule. Vector joining two points = head minus tail. Points are collinear if BC⃗ = λAB⃗ and the two vectors share a point.
  3. 3 · Find AB⃗ and BC⃗.
    • AB⃗ = (4 − 2)î + (4 − 1)ĵ + (1 + 1)k̂ = 2î + 3ĵ + 2k̂
    • BC⃗ = (8 − 4)î + (10 − 4)ĵ + (5 − 1)k̂ = 4î + 6ĵ + 4k̂
  4. 4 · Compare them. BC⃗ = 2AB⃗, so they are parallel. They share the point B, so A, B, C are collinear.
  5. 5 · Find the lengths.
    • |AB⃗| = √(4 + 9 + 4) = √17
    • |BC⃗| = 2√17
    • AC⃗ = 6î + 9ĵ + 6k̂, so |AC⃗| = √(36 + 81 + 36) = √153 = 3√17
  6. 6 · Check and box the answer. |AB⃗| + |BC⃗| = √17 + 2√17 = 3√17 = |AC⃗|. A, B, C are collinear, with B between A and C.

Scoring tip: Always mention the common point. Two parallel vectors with no shared point could lie on two different parallel lines.