Question

How is (A+B) X (A-B) vector is equal to 2(A X B)

Answer 1

Answer: You can assume as vector sign -> above a and b

While answering this question, I will assume you know what a cross product between two vectors is.

The question basically uses the distributive law. Solving it systematically:

(a⃗ +b⃗ )x(a⃗ −b⃗ )

=a⃗ x(a⃗ −b⃗ )+b⃗ x(a⃗ −b⃗ ) (Using the left distributive law)

=a⃗ xa⃗ −a⃗ xb⃗ +b⃗ xa⃗ −b⃗ xb⃗ (Opening the brackets)

Now, the cross product of a vector with itself is the null vector, which you should know. So, a⃗ xa⃗ =0⃗ and b⃗ xb⃗ =0⃗

Also, the fundamental anti-commutative law of cross products states: a⃗ xb⃗ =−b⃗ xa⃗

This is really easy to prove, and can be visualized easily if you know how a cross-product is depicted in space.

Using all the above results, we have:

a⃗ xa⃗ −a⃗ xb⃗ +b⃗ xa⃗ −b⃗ xb⃗ =0⃗ −(−b⃗ xa⃗ )+b⃗ xa⃗ −0⃗ =2(b⃗ xa⃗ )=−2(a⃗ xb⃗ )

This is the final result.

Hope this helps