Can my rich father take zakaah and spend it on my brother so that he can complete his studies?

Praise be to Allah.

Firstly: 

The fuqaha’ are unanimously agreed that the father is not obliged to spend on his adult son if he is able to earn a living. For more information please see the answer to question no. 13464 

Because spending on an adult son who is able to earn a living is not obligatory for the father, it is permissible for the son to take zakaah as much as will suffice his needs, even if he is a student of non-religious knowledge (as opposed to Islamic knowledge). For more information please see the answer to question no. 95418. 

In this case there is nothing wrong with delegating his father to take it on his behalf, then give it to him to help him complete his studies. 

The criteria for it being permissible to take zakaah is if the student of non-religious knowledge is not able to combine work and study, or if combining them will affect his ability to study. In that case there is nothing wrong with him taking zakaah money as much as will suffice his needs. But if he is able to combine them both without suffering any harm, then it is not permissible for him to take zakaah money, because of the report narrated by Hishaam from his father from ‘Ubaydullah ibn ‘Adiyy who said: Two men told me that they came to the Prophet (blessings and peace of Allah be upon him) during the Farewell Pilgrimage to ask him for charity. The Messenger of Allah (blessings and peace of Allah be upon him) looked them up and down and saw that they were healthy young men. He said: If you wish I will give you some, but there is no share of it for one who is independent of means or one who is strong and able to earn a living.” Narrated by Imam Ahmad, 21985; Abu Dawood, 1633; classed as saheeh by al-Albaani (may Allah have mercy on him) in Saheeh Abi Dawood, no. 1443 

There is no doubt that it is better and is preferable for his father to help him from his own wealth or for him to strive to combine work and study as much as he can, so as to avoid going against those scholars who said that this is not allowed and so as to refrain from taking what is in people’s hands, because the Prophet (blessings and peace of Allah be upon him) said: “The upper hand (the one that gives) is better than the lower hand (the one that receives).” Narrated by al-Bukhaari, 1429; Muslim, 1715 

And he (blessings and peace of Allah be upon him) said concerning zakaah wealth: “It is the dregs of the people.” Narrated by Muslim, 1784 

It was narrated from al-Miqdaam (may Allah be pleased with him) that the Messenger of Allah (blessings and peace of Allah be upon him) said: “No one ever eats any food better than that which he earns with the work of his own hand. The Prophet of Allah Dawood (peace be upon him) used to eat from that which he earned with his own hand.

Narrated by al-Bukhaari, 2072 

It was narrated from Rifaa‘ah ibn Raafi‘ (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) was asked: Which earning is best? He said: “(That which earned) by the work of a man’s own hand and every honest transaction.” Narrated by Imam Ahmad, 16628; al-Bazzaar, 3731. Classed as saheeh by Shaykh al-Albaani (may Allah have mercy on him) in as-Silsilah as-Saheehah, 2/106 

And Allah knows best.