Prove Function G Is Linear First of all $v, w, z$ are in $\mathbb R^3$ as vectors, and there are functions $G(v, w, z) = (v \times  w) \times z$, this is cross product. So I know this function is linear in the 3 arguments, so for example $G(x+ y)= G(x)+G(y)$ and $G(ax)=aG(x)$, and $a$ is a constant. First of all what is means to be linear in second argument? And, also, prove this $G$ function is linear in the second argument.

A function of multiple vectors can be said to be linear or not with respect to each of its arguments (whereas a function of one vector is said to be linear or not).
G is linear in its second argument if $G(v,aw_1+bw_2,z)=aG(v,w_1,z)+bG(v,w_2,z)$.
The left hand side is $(v \times(aw_1+bw_2))\times z=(v\times aw_1+v\times bw_2)\times z=(v\times aw_1)\times z+(v\times bw_2) \times z$, which equals the right hand side.
