If a⃗ is any non-zero vector, then (a⃗·i⃗)i⃗ + (a⃗·j⃗)j⃗ + (a⃗·k⃗)k⃗ =

Explanation

Write a⃗ = aₓi⃗ + aᵧj⃗ + a_zk⃗. Then (a⃗·i⃗)i⃗ = aₓi⃗, (a⃗·j⃗)j⃗ = aᵧj⃗, and (a⃗·k⃗)k⃗ = a_zk⃗. Adding the three components reconstructs a⃗.