Given a line L in three-dimensional space and a point P, how can we find the normal vector of L under the constraint that the normal passes through P?
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Let the line and point have position vectors $\vec r=\vec a+\lambda \vec b$ ($\lambda$ is real) and $\vec p$ respectively. Set $(\vec r-\vec p).\vec b=0$ and solve for $\lambda$ to obtain $\lambda_0$. The normal vector is simply $\vec a+\lambda_0 \vec b-\vec p$.