Learn Russian: Вы можете разрушить свою жизнь ради богатства и тщеславия. - You can destroy your life for the sake of wealth and vanity.

You could, but it changes the nuance. Both take the Genitive case, but ради specifically means for the sake of and often implies making a sacrifice or having a very strong, high-stakes motivation. для is a much more neutral for. Because destroying your life is a massive sacrifice, ради is the perfect fit here.

What case are the words богатства and тщеславия, and what are their dictionary forms?

They are in the Genitive case, which is strictly required after the preposition ради. Their nominative (dictionary) forms are богатство (wealth) and тщеславие (vanity). Since both are neuter nouns, their endings change from -о to -а, and from -е to -я in the Genitive.

Why does the sentence use свою rather than вашу for your?

Whenever the object belongs to the subject of the sentence, Russian strongly prefers the reflexive possessive pronoun свой (in all its gendered and case forms). Since the subject вы is doing the action to their own life, свою is much more natural and grammatically correct than вашу.

Why is the perfective verb разрушить used here instead of the imperfective разрушать?