Suppose $u,v$ are $2$ vectors in a real inner product space $(V, \langle \cdot , \cdot \rangle )$. write each of the expressions, $(u-5v,3u-2v)$, $(2u+7u,u+3(u-5v,3u-2v)v)$ in terms of $(u,v)$, $\| u \|^2$, and $\| v \|^2$
heres my take so far:
i think i have to use axioms such as $(u,v)$=$(v,u)$ or $(u,v+w)$=$(u,v)$+$(u,w)$ and if i do that i get $(u,3u-2v-5v)$+$(-5v+3u-2v)$? i am very confused.
I'll do the first one for you. The second is done in exactly the same way.
Here I'll use your two axioms, plus one more: $k\langle x,y\rangle = \langle kx, y\rangle = \langle x, ky\rangle$ where $k\in \Bbb R$ and $x,y\in V$.
$$\require{enclose}\begin{align}\langle u-5v,3u-2v\rangle &= \langle u-5v, 3u\rangle + \langle u-5v, -2v\rangle \\ &= 3\langle u-5v, u\rangle - 2\langle u-5v, v\rangle \\ &= 3\langle u, u-5v\rangle - 2\langle v, u-5v\rangle \\ &= 3\langle u, u\rangle + 3\langle u,-5v\rangle - 2\langle v, u\rangle - 2\langle v,-5v\rangle \\ &= 3\| u\|^2 - 15\langle u,v\rangle - 2\langle u, v\rangle + 10\langle v,v\rangle \\ &= \enclose{box}{3\| u\|^2 - 17\langle u,v\rangle + 10\| v\|^2}\end{align}$$
Go through and make sure you understand which axioms I used at each step.