Good one guys! I've been able to prove (a) and (b), but (c) just got me struggling for a week now, and when I asked my orientator for help he said that I had to prove the parallelpiped diagonal formula using vector arguments such as cos, sin, and vector product. And the the relation between the short diagonal and long diagonal.
Me and a couple of friends tried to brainstorm how we could interpret the problem but It just got messy.
We tried to draw it a bit of different ways but we could not make sense.
Anyone willing to help we would be very grateful.
If a parallelogram is defined by $\vec{u}$ and $\vec{v}$, then $\vec{u} + \vec{v}$ and $\vec{u} - \vec{v}$ are the two diagonals of the parallelogram.
Therefore, the second one shows that the sum of the squares of all four sides of the parallelogram (because a parallelogram has two pairs of equal sides, we have a factor of $2$) is equal to the sum of the squares of its diagonals - a well known theorem. With this in mind, try part (a)!