向く and 向ける

向く and 向ける are a intransitive and transitive pair of verbs meaning “facing”.  Originally, they referred to physically facing something:

• 道路に向いている窓
  A window facing the road
  
  
• カメラを海に向けた
  Faced the camera towards the ocean
  
  
• 刀を敵に向けた
  Turned the sword towards the enemy. 
  
  

However, they may also be used to indicate suitability or purpose.

向く

向く
Suited to

向く is an intransitive verb meaning “suited to”. It is used to describable suitability. 
It tends to be in the progressive form. 
This verb uses に、due to it being a direction (In Japanese). 

Examples

• 医者に向いていない人
  A person not suited to becoming a doctor
  
  
• 子供が好きな彼は幼稚園の先生に向いている
  Since he likes children, he’s suited to be a kindergarten teacher.
  
  
• リズムがない人は音楽家に向いていない
  People with no rhythm aren’t suited to be musicians. 
  
  

向き

Suited for

向き is the ます stem of 向く, and may attach to the ます stems of verbs or nouns. 
Like its original verb, it is used to mean something is suited for something else.

Examples

• 子供向きの本
  A book suitable for children.
   
• 医者向きのウエブサイト
  A website suitable for doctors
  
  

向ける

向ける
Meant for

向ける is a transitive verb meaning “looking at”. It used to describe goals or purpose.
The action after 向ける is meant for the word before 向ける。 
向ける also uses に. 

Examples

• 明日のテストに向けて今夜は勉強する
  Looking at the test tomorrow, I’m going to study tonight. 
  
  
• 将来に向けて勉強すべきだ
  Looking towards the future, you should study.
  
  
• 学生に向けてウエブサイトをつくる
  Looking at students, make a website. (meant for students). 
   

向け

Meant for

Like 向き、向け is the ます stem of 向ける; and may attach to other ます stem verbs or nouns.
向け describes goal or purpose. You can translate it as meant. 
向け is more restrictive than 向き, as it describes the original intention. 

Examples

• 子供向けの本
  A book meant for children. 
  
  
• 外国人向けのウエブサイト
  A website meant for foreigners.
  
  
• 20代の人向けのアプリ
  An app meant for 20 year olds. 
  
  

向き and 向け Comparison

Here is a quick comparison to make the difference more clear:

• 子供向きの本 
  A book suitable for children.
  
  
• 子供向けの本
  A book meant for children.
  
  

One more example pulled from stack exchange:

• 犬向きの公園
  A park suitable for dogs. I.e. open spaces, grass, trees
  .
• 犬向けの公園
  A park meant for dogs. I.e. a dog park. Mostly for dogs.