Problem of understanding a vector calculus formula

Question

I have encountered the following formula during my course in electromagnetic field theory: $$ {\displaystyle {\vec {\nabla }}\times \varphi {\vec {A}}=\varphi {\vec {\nabla }}\times {\vec {A}}-{\vec {A}}\times {\vec {\nabla }}\varphi .} $$ I just do not understand the menaing of the first part in front of the minus sign. I have also seen a similar situation in the following formula: $$ {\displaystyle {\vec {\nabla }}\times ({\vec {A}}\times {\vec {B}})=({\vec {B}}\cdot {\vec {\nabla }}){\vec {A}}-{\vec {B}}({\vec {\nabla }}\cdot {\vec {A}})+{\vec {A}}({\vec {\nabla }}\cdot {\vec {B}})-({\vec {A}}\cdot {\vec {\nabla }}){\vec {B}}.} $$ But now with a Vectorfield and not with a scalar function. Can somebody please help me with this lack of understanding?