Tangent planes to quadrics

In the tutorials, we revisited that for a curve that satisfies $\langle \gamma , \gamma \rangle = c$ one always has $\langle \gamma’ , \gamma \rangle = 0$, i.e., $\gamma \perp \gamma’$ with respect to the product $\langle \cdot , \cdot \rangle$.

This means that tangent vectors $v$ to the quadric $Q = \left\{ x \mid \langle x , x \rangle = c \right\}$ at a point $p \in Q$ are characterized by $\langle p , v \rangle = 0 \Leftrightarrow v \in p^\perp$. However, the affine plane that is tangent to $Q$ at $p$ is given by $\left\{ x = p + v \mid v \in p^\perp \right\} = \left\{ x \mid \langle x , p \rangle = \langle p , p \rangle = c \right\}$.

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