**Exercise:**

Assume that 4 vectors in \(\mathbb{R}^3\) satisfy \(A+B+C+D\)=0.

Simplify

$$ A\times B-B\times C+C \times D-D\times A$$

** Solution: ** By the anti-symmetry \(-B\times C = C\times B\)

and \(-D\times A = A\times D\) we can rewrite the expression as

$$ A\times B+C\times B+C \times D+A\times D$$

Using the distributive law of the cross product, we have

$$ (A+C)\times B+(C+A) \times D$$

and again

$$ (A+C)\times (B+D) =** $$

However, as \(F=(A+C) = -(B+D)\) we get

$$ **= F\times (-F) = 0 $$

**Exercise:** Integrate \(dx/y\) over the circle (oriented counterclockwise)

\(C:=\{x^2+y^2=R^2\}\).…