I am looking at some past exam practice here, and came across (8 on the worksheet):
Express the dot product of two vectors $a \cdot b$ through the lengths
$M$ and $N$ of their sum and difference: $M = |a + b|$ and $N = |a – b|$.
My initial thought: Consider the dot product of $(a+b) \cdot (a+b)$ and expand:
$$(a+b) \cdot (a+b) = a^2 + 2a \cdot b + b^2$$
Now I have a connection between $2a \cdot b$ and $(a + b) \cdot (a + b)$ that I can hopefully use. So I tried solving for $2a \cdot b$:
$$(a + b) \cdot (a + b) – (a^2 + b^2) = 2a \cdot b$$
This is where I get stuck, Something that looked relatively promising involved using complex numbers:
$$(a + b) \cdot (a + b) – (a + ib) \cdot (a – ib) = 2 a \cdot b$$
So we have now two terms that almost look like $a + b$ and $a – b$, but I'm not really sure where to go from here.
Best Answer
Hint: You started out with the right idea but took it in a wrong direction. How does $M$ relate to $(a+b)\cdot(a+b)$? Does that suggest something you might do with $(a-b)$?