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

Jun 2017
345
6
Lima, Peru
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
Sep 2015
2,885
1,609
USA
$(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)
$
 

skipjack

Forum Staff
Dec 2006
21,387
2,410
It's convenient to rotate the axes about the $z$-axis first, so that you can use $3\hat {i}$ for $\vec{A}$, etc.