I know that if we think of u and v as two vectors in R^3, the magnitude of the dot-product of u and v is |u|v|cos(x), where x is the angle between them. If a set of vectors is linearly dependent, at least one is a multiple of the other which means (I think) they don't point in the same direction. This means (again, I think) that the range of u * v is R < |1|.
I just have two questions:
I don't know if that is the answer, my teacher explained it and this is how I interpreted it, but I'm not sure it is correct. So, is this the answer?
If it is the answer, how would I write that in interval notation? I learned interval notation a while back, but I forgot.
$u.v$ lies between $-1$ and $+1$. The required interval is $[-1,1)$. You also have to show that for any $t \in [-1,1)$ there exist linearly independent unit vectors $u$ and $v$ such that $u.v=t$. For this take $u=(1,0,0)$ and $v =(t,\sqrt {1-t^{2}},0)$. [You cannot have linearly independent unit vectors $u$ and $v$ such that $u.v=1$. This is a consequence of the condition for equality in Cauchy -Schwarz inequlity].