Let $V = \mathbb{R}^3$ and let $H$ be the subset of $V$ of all points on the plane $7 x + 4 y + 6 z = 0$. Is $H$ a subspace of the vector space $V$?

1. Does $H$ contain the zero vector of $V$?
   choose H contains the zero vector of V H does not contain the zero vector of V
   
   
2. Is $H$ closed under addition? If it is, enter CLOSED. If it is not, enter two vectors in $H$ whose sum is not in $H$, using a comma separated list and syntax such as $\verb+<1,2,3>, <4,5,6>+$.
   
   
   
3. Is $H$ closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in $\mathbb{R}$ and a vector in $H$ whose product is not in $H$, using a comma separated list and syntax such as $\verb+2, <3,4,5>+$.
   
   
   
4. Is $H$ a subspace of the vector space $V$? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3.
   choose H is a subspace of V H is not a subspace of V