Let a⃗ = 2î − ĵ + 2k̂. (i) Find the unit vector along a⃗. (ii) Find a vector of magnitude 9 along a⃗. (iii) Show that b⃗ = −4î + 2ĵ − 4k̂ is collinear with a⃗ and say whether it has the same direction.

1. 1 · Write what is given and what is to be found. Given: a⃗ = 2î − ĵ + 2k̂, b⃗ = −4î + 2ĵ − 4k̂. To find: â, a vector of length 9 along a⃗, and whether b⃗ is collinear with a⃗.
2. 2 · Write the rules. â = a⃗/|a⃗|. A vector of length k along a⃗ is kâ. b⃗ is collinear with a⃗ if b⃗ = λa⃗.
3. 3 · Find the magnitude and the unit vector.
   • |a⃗| = √(4 + 1 + 4) = 3
   • â = (2î − ĵ + 2k̂)/3
4. 4 · Find the vector of magnitude 9. 9â = 3(2î − ĵ + 2k̂) = 6î − 3ĵ + 6k̂
5. 5 · Test for collinearity. b⃗ = −4î + 2ĵ − 4k̂ = −2(2î − ĵ + 2k̂) = −2a⃗. So b⃗ is collinear with a⃗. λ = −2 is negative, so b⃗ points the opposite way.
6. 6 · Check and box the answer. â = (2î − ĵ + 2k̂)/3; 6î − 3ĵ + 6k̂; b⃗ = −2a⃗, collinear and opposite in direction.

Scoring tip: For collinearity, show the scalar λ explicitly. 'They look parallel' earns nothing.