Rotate a Vector3 direction

Right, though Unity just decides to only implement Quaternion * Vector multiplication. Mathematically in order to rotate a vector you have to multiply the quaternion on the left side of the vector and the complex conjugate on the right side which is all implicit in that one multiplication operator in Unity. To see an actual run-down I always recommend this Numberphile video. It does simplify the concept a lot, but you get a clue what's actually going on. I once manually derived the quaternion - vector multiplication myself by literally expanding and then simplify the following

 (q.w + q.x*i + q.y*j + q.z*k) * (v.x*i + v.y*j + v.z*k) * (q.w - q.x*i - q.y*j - q.z*k)

though always remember the rules i² = j² = k³ = ijk = -1 and the order. So i*j == k but j*i == -k. i,j and k form a "ring". Writing them in a circle clockwise and you can derive the sign easily.

 // clockwise
 i*j = k
 j*k = i
 k*i = j
 // counter clockwise
 k*j = -i
 j*i = -k
 i*k = -j

in the end the real parts should cancel and you're left with just i,j and k which is your rotated vector.

Here's my run-down. The resulting equations essentially form a 3x3 matrix.