If A×B = C ,then which of the following s

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 $(\overrightarrow A + \overrightarrow B )$vector also, must lie in the plane formed by vector $\overrightarrow A $ and $\overrightarrow B $. Thus $\overrightarrow C $ must be perpendicular to $(\overrightarrow A + \overrightarrow B )$ also but the cross product $(\overrightarrow A \times \overrightarrow B )$ gives a vector $\overrightarrow C $ which can not be perpendicular to itself. Thus the last statement is wrong.