If $\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{C}$, then which of the following statements is wrong

From the property of vector product, we notice that $\overrightarrow{C}$ must be perpendicular to the plane formed by vector $\overrightarrow{A}$ and $\overrightarrow{B}$. Thus $\overrightarrow{C}$ is perpendicular to both $\overrightarrow{A}$ and $\overrightarrow{B}$ and $\left(\overrightarrow{A}+\overrightarrow{B}\right)$ vector also must lie in the plane formed by vector $\overrightarrow{A}$ and $\overrightarrow{B}$ . Thus $\overrightarrow{C}$ must be perpendicular to $\left(\overrightarrow{A}+\overrightarrow{B}\right)$ but the cross product $\left(\overrightarrow{A}\times \overrightarrow{B}\right)$ gives a vector $\overrightarrow{C}$ which cannot be  perpendicular to itself. Thus the last statement is wrong.