What exactly is the difference between $\vec{a} $ and $\lVert\vec{a}\rVert$ ?
I know $\lVert\vec{a}\rVert$ is the magnitude of a vector $\vec{a} $ , but what is $\vec{a} $ then exactly? I know it's a vector, but how can you represent it? Let's say $\vec{a} $ is a vector in $\mathbb{R}^2$, if you draw $\vec{a} $ in $\mathbb{R}^2$, don't you automatically draw $\lVert\vec{a}\rVert$ ?
Also, what is the difference between $ \lVert\vec{a}\rVert \cdot \big\lVert\vec{b}\big\rVert $ and $\vec{a} \cdot \vec{b} $? To me it seems like multiplying the same, but I know it isn't.