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How can I use the minimum distance between two skew lines to find the exact coordinate where the distance between the two lines is the shortest? Let's say the lines have the equations:

$$g = (1,3,5)^T + b(7, 11, 13), b \in \mathbb{R}$$

$$h = (2,7,5)^T + c(1,2,5), c \in \mathbb{R}$$

I used an online calculator to find the minimum distance to be $1.615$, however I'm not sure how I can use to figure out the coordinates where the distance is the shortest. I assume I have to first calculate the values of $b$ and $c$ where the distance is the shortest and then use that to simply the two equations, which will give me the coordinates - but I don't know how to do that.

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  • $\begingroup$ You can minimize $|g-h|^2$ in $b$ and $c$. $\endgroup$
    – A.Γ.
    Nov 30, 2019 at 22:12

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Hint:

It is given by the common perpendicular of the two lines. If $$\vec u and $\vec v$ are the directing vectors of the lines, a directing vector of the common perpendicular is their cross-product.

You can determine the line passing though a point $(a,b,c)$ with directing vector $\vec u\times\vec v$ which meets both lines: the compatibility conditions of the resulting equations will yield the values of $a, b,c$ and the values of the parameters $s$ and $t$.

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