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

Question

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

$\vec{A}=\left \langle 3\cos 53^{\circ}, 3 \sin 53^{\circ} \right \rangle$

$\vec{B}=\vec{A}=\left \langle 3\cos (53^{\circ}+30^{\circ}), 3 \sin (53^{\circ}+30^{\circ}) \right \rangle$

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

Can someone help me with this?.

Answer 1

Hint. By expanding the cross product we find $$(\vec{A}+\vec{B})\times(\vec{A}-\vec{B})=\vec{A}\times\vec{A}+\vec{B}\times\vec{A}-\vec{A}\times\vec{B}-\vec{B}\times\vec{B}.$$ Are you able to find each of the 4 cross-products on the right-hand side?

Recall the algebraic properties of the cross product!

Answer 2

The answer is $-9\vec k$. In fact, the angle between $\vec A$ and $\vec B$ has $30^\circ$ degrees. On the other hand\begin{align}\left(\vec A+\vec B\right)\times\left(\vec A-\vec B\right)&=\overbrace{\vec A\times\vec A}^{\phantom{0}=0}+\overbrace{\vec B\times\vec A}^{\phantom{-\vec A\times\vec B}=-\vec A\times\vec B}-\vec A\times\vec B-\overbrace{\vec B\times\vec B}^{\phantom{0}=0}\\&=-2\vec A\times\vec B.\end{align}The length of $\vec A\times\vec B$ is $3\times3\times\sin(30^\circ)=\frac92$, and therefore the answer is $-9\vec k$; in order to see why it is this and not $9\vec k$, use the right-hand rule.

Answer 3

The cross product is associative and anti-commutative. (A+ B)X(A- B)= AXA- AXB+ BXA- BXB= -AXB- AXB= -2AXB. Yes, since both A and B lie in the xy-plane -AXB is in the negative z direction. And since $|AXB|= |A||B| sin(\theta)$, here |AXB|= 2(3)(3) sin(150)= 9. (A+ B)X(A- B)= -9k