Why will this lookVector not work? It will not make it face the same way character is facing.

local plr = game.Players.LocalPlayer
local mouse = plr:GetMouse()

mouse.KeyDown:connect(function(key)
	if key == "f" then
		local part = Instance.new("Part")
		local fire = Instance.new("Fire",part)
		local Mesh = Instance.new("SpecialMesh",part)
		Mesh.MeshId = "rbxassetid://475201433"
		Mesh.Scale = Vector3.new(0.1,0.1,0.1)
--		Mesh.TextureID = "rbxassetid://475201512"
		part.Parent = workspace
		part.Name = 'FireBall'
		part.CanCollide = false
		part.Anchored = true
		part.Position = plr.Character.Torso.Position
		part.CFrame.lookVector = CFrame.new(plr.Character.Head.CFrame.lookVector) -- Problem here
	end
end)

Error: 20:58:21.502 - lookVector cannot be assigned to

It makes the part but it will not face the way the player is facing.

As the error you shared indicates, you cannot directly set the lookVector of a Part. You can, however, set the Position and the CFrame properties.

You can set the CFrame rotation of the Part to the direction of the player's head like this:

head_cframe = plr.Character.Head

-- remove the positional component from the CFrame, left with rotation only
head_cframe_rotation = head_cframe - head_cframe.p

-- Add the rotational and positional components together 
part.CFrame = head_cframe_rotation + part.Position