How can I show that these two formulas for $\langle v,w \rangle$ might define inner products on $\Bbb R^2$?
1.) $v_1 w_2 + v_2 w_1$
2.) $2v_1 w_1 + (v_1-v_2)(w_1-w_2)$
I know that for number 1 it does not define an inner product and for number 2 it does define an inner product but why? I also know in order to define an inner product it must satisfy the conditions of bilinearity, symmetry, and positivity but I am confused on how I can show that?