I've been studying from Gilbert Strang's 1993 Linear Algebra text, and right in the first chapter, I've found a problem on dot products whose answer in the back is tough for me to make sense out of...
Question:
Pick any numbers that add to x + y + z = 0. Find the angle between your vector v = (x, y, z) and the vector w = (z, x, y). Challenge Question: Explain why (v · w) / ||v||||w|| is always -1/2.
The former part is easy. I found cos(angle) = -1/2 using v = (-1, -1, 2).
Answer given for the latter part:
...Write v · w = xz + yz + xy as (1/2)(x + y + z)2 - (1/2)(x2 + y2 + z2) whis is (-1/2)(x2 + y2 + z2).
I tried rewriting it as suggested, which could make sense of that long middle term. It's is the final phrase "which is.." that throws me off.
Help so appreciated!
Noting that $v\cdot w = xz+yz+xy = \frac{1}{2}(x+y+z)^2 - \frac{1}{2}(x^2+y^2+z^2)$ we then have $$\frac{v\cdot w}{\|v\| \|w\|}=\frac{\frac{1}{2}(x+y+z)^2 - \frac{1}{2}(x^2+y^2+z^2)}{x^2+y^2+z^2}=\frac{0-\frac{1}{2}(x^2+y^2+z^2)}{(x^2+y^2+z^2)}=-\frac{1}{2}.$$