Suppose I have a vector say v1=(1,2,3) and the dot product of another vector(v2) with v1 is zero. What other information do we need too find v2. I thought this seemed like a pretty trivial question however when I went to do it I had difficulty.
I know that a vector is a quantity with both magnitude and direction. So in other to get the vector v2 we would need its direction. due to the fact that you can have multiple different vectors prependicular. You would also need its magnitude. So is it the case that the dot product gives us almost no information in findng the vector v2,
In general the fact that $v_1\cdot v_2 = 0$ means that $v_1$ and $v_2$ are orthogonal. In your case, you are working in $\mathbb{R}^3$, so that the set of vectors orthogonal to $v_1$ forms a plane (if you want to visualize this, think of $v_1$ as being the vector $\langle 1, 0, 0\rangle$ and think about the vectors perpendicular to this one). So you will need a lot more information than just length to figure out what $v_2$ is.
Even in $\mathbb{R}^2$, the set of vectors orthogonal to $v_1$ forms a line, so there are still an infinite number of them.