# How can I find the cross product of an inner sum and difference between two vectors?

#### Chemist116

The problem is as follows:

The figure from below shows vectors $\vec{A}$ and $\vec{B}$. It is known that $A=B=3$. Find $\vec{E}=(\vec{A}+\vec{B})\times(\vec{A}-\vec{B})$. The alternatives are:

$\begin{array}{ll} 1.&-18\hat{k}\\ 2.&-9\hat{k}\\ 3.&-\sqrt{3}\hat{k}\\ 4.&3\sqrt{3}\hat{k}\\ 5.&9\hat{k}\\ \end{array}$

What I've attempted here was to try to decompose each vectors

A⃗ =⟨3cos53∘,3sin53∘⟩A→=⟨3cos⁡53∘,3sin⁡53∘⟩

B⃗ =A⃗ =⟨3cos(53∘+30∘),3sin(53∘+30∘)⟩B→=A→=⟨3cos⁡(53∘+30∘),3sin⁡(53∘+30∘)⟩

But attempting to use these relationships does seem to extend the algebra too much. Does there exist another way? Or could it be that I am overlooking something?

Can someone help me with this?

Last edited:

#### romsek

Math Team
$(A+B)\times (A-B) = \\ A \times (A-B) + B \times (A-B) = \\ A \times A - A \times B + B \times A - B \times B = \\ 0 - A \times B + B \times A - 0 =\\ 2 (B \times A)$

• Chemist116 and topsquark

#### skipjack

Forum Staff
It's convenient to rotate the axes about the $z$-axis first, so that you can use $3\hat {i}$ for $\vec{A}$, etc.