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

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$\vec{C} ⊥ \vec{A}$

$\vec{C} ⊥ \vec{B}$

$\vec{C} ⊥ (\vec{A} + \vec{B})$

$\vec{C} ⊥ (\vec{A} \times \vec{B})$

answer is D.

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Detailed Solution

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