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Direction cosines and direction ratios of a vector sheet
3-mark answer frame

For r⃗ = 6î − 2ĵ − 3k̂: (i) find its direction cosines, (ii) verify that l² + m² + n² = 1, and (iii) find a vector of magnitude 14 in the direction of r⃗.

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: r⃗ = 6î − 2ĵ − 3k̂. To find: l, m, n; check the identity; a vector of length 14 along r⃗.
  2. 2 · Write the formulas. l = a/|r⃗|, m = b/|r⃗|, n = c/|r⃗|; r̂ = lî + mĵ + nk̂.
  3. 3 · Find the magnitude. |r⃗| = √(36 + 4 + 9) = √49 = 7
  4. 4 · Write the direction cosines. l = 6/7, m = −2/7, n = −3/7
  5. 5 · Verify the identity. 36/49 + 4/49 + 9/49 = 49/49 = 1
  6. 6 · Find the vector of magnitude 14. 14r̂ = 14(6î − 2ĵ − 3k̂)/7 = 12î − 4ĵ − 6k̂
  7. 7 · Check and box the answer. l, m, n = 6/7, −2/7, −3/7; the identity holds; required vector 12î − 4ĵ − 6k̂ (its length is 2 × 7 = 14).

Scoring tip: Keep the signs of the components in the direction cosines. A lost minus sign is the most common error here.