# Inverse Kinematics with SE(3) Target Derivation

Why are there square brackets around [V_b] when deriving it, but not when using it in the update?

The vertical pipes denote the magnitude. It looks like the magnitude of the velocity vector must be above a threshold value (epsilon) for the robot to be considered in motion. The square brackets denote vector/matrix. Then, in the equation denoted with 1., you are multiplying a matrix with a vector, so no brackets or pipes are necessary.
Your question lacks sufficient information to provide a precise answer. It seems that $$\mathcal{V}_b$$ is a vector in $$\mathbb{R}^3$$ and is used in combination with the Jacobian of an "end-effector" $$b$$.
I believe that the braked notation around $$\mathcal{V}_b$$ means: "Take the components of the right-hand side and assign them to the components of the left-hand side." The author likely uses this notation because the logarithm of an orthogonal 3x3 matrix results in a skew-symmetric 3x3 matrix, which has only 3 degrees of freedom.
The slides you mentioned present a less common formulation of inverse kinematics, acknowledging the manifold structure of $$SO(3)$$ and utilizing the quasi-Newton algorithm on the tangent space. The brackets are likely used to represent elements of the tangent space of $$SO(3)$$ in $$\mathbb{R}^3$$ (as they are isomorphic).