The vectors $p, q, r$ and $s$ satisfy the identities $p+q+r+s = 0$, and $\|p\| = \|q\| = \|r\| = \|s\|$. Prove that $p\cdot q = r\cdot s$ I got asked this question in class today and I am unsure on where to start!
The vectors $p, q, r$ and $s$ satisfy the identities $p+q+r+s = 0$, and $\|p\| = \|q\| = \|r\| = \|s\|$.
Prove that $p\cdot q = r\cdot s $.
Rewrite the formula to get $$p+q = -r-s$$
Then compute the norm of this sum $$\|p\|^2+2p\cdot q+\|q\|^2=\|r\|^2+2r\cdot s+\|s\|^2$$
cancelling the norms (which are all equal) from both sides gives $$2p\cdot q=2r\cdot s$$
We have $$|p+q|^2 =|r+s|^2 $$
$$|p|^2 + 2(p\cdot q) +|q|^2 =|r|^2 + 2(r\cdot s) +|s|^2 $$
$$|p|^2 + 2(p\cdot q) +|q|^2 =|p|^2 + 2(r\cdot s) +|q|^2 $$