Let a⃗ = î + 2ĵ − k̂, b⃗ = 2î − ĵ + 3k̂ and c⃗ = 3î + ĵ + k̂. Find λ such that a⃗ + λb⃗ is perpendicular to c⃗. Also find the projection of a⃗ on c⃗.

1. 1 · Write what is given and what is to be found. Given: a⃗, b⃗, c⃗. To find: λ with (a⃗ + λb⃗) ⊥ c⃗, and the projection of a⃗ on c⃗.
2. 2 · Write the rules. Perpendicular vectors have dot product 0. Projection of a⃗ on c⃗ = (a⃗ · c⃗)/|c⃗|.
3. 3 · Write a⃗ + λb⃗ in components. a⃗ + λb⃗ = (1 + 2λ)î + (2 − λ)ĵ + (−1 + 3λ)k̂
4. 4 · Set the dot product to zero.
   • 3(1 + 2λ) + 1(2 − λ) + 1(−1 + 3λ) = 0
   • 3 + 6λ + 2 − λ − 1 + 3λ = 0, so 4 + 8λ = 0
   • λ = −1/2
5. 5 · Find the projection. a⃗ · c⃗ = 3 + 2 − 1 = 4 and |c⃗| = √(9 + 1 + 1) = √11. Projection = 4/√11.
6. 6 · Check and box the answer. λ = −1/2: a⃗ − (1/2)b⃗ = 0î + (5/2)ĵ − (5/2)k̂, and its dot with c⃗ is 0 + 5/2 − 5/2 = 0. Projection = 4/√11.

Scoring tip: Expand the dot product term by term on one line. Most mistakes come from dropping a sign inside a bracket.