Rotate a Vector by Quaternions

I'm trying how to work out how to Rotate a Vertex using Quaternions, using a scientific calculator, or on paper. Exam preparation.

My lecturer has given us this; Quaternion = (-0.5, 0, -0.7071067, 0.5) Vertex = (23, 10, 18)

The way it's been explained to us is like this;

• We have a vertex called p

• We have a quaternion called q

• We store p within a quaternions vector component, we'll call this K

• K = (0, p)

• Finally we do the normal quaternion multiplication

• p' = qKq-1

I'm just trying to work out how I break this down so it's easier to understand so I am able to find out the result. I know how to do quaternion multiplication, but it seems confusing as I only have one w component in the quaternion, and only the x, y,z in the vector.

• What happens when you try to do , say, the left multiplication, (q.w, q.x, q.y, q.z) * (0, p.x, p.y, p.z), using your rules for the quaternion basis units i*i = -1, i*j = k... etc? – DMGregory May 2 '19 at 13:43
• Maybe you'd find it easier to find answers on Mathematics StackExchange ? – TomTsagk May 2 '19 at 14:01