For two vectors $\vec{u}$ and $\vec{v}$, define

$\vec{u} * \vec{v}=\vec{u} \times(\vec{u} \times \vec{v})+\vec{v} \times(\vec{v} \times \vec{u}) .$

Then

• A

  $\vec{u} *(\vec{u}+\vec{v})=\vec{u} * \vec{v}$, for any two arbitrary vectors $\vec{u}$ and $\vec{v}$
• Correct answer
  B

  $\vec{u} * \vec{v}=\vec{v} * \vec{u}$ and $\vec{u} * \vec{u}=\overrightarrow{0}$
• Correct answer
  C

  the angle between the vectors $\vec{u}-\vec{v}$ and $\vec{u} * \vec{v}$ is $90^{\circ}$, where $\vec{u}$ and $\vec{v}$ are two distinct non-zero vectors
• D

  $\vec{u} * \vec{v} \neq \overrightarrow{0}$ if $\vec{u}$ and $\vec{v}$ are distinct non-zero vectors

Hence, the correct answer is options: 'B', 'C'.