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I would like to ask if it's possible to calculate the shortest distance in space between a point and a straight line in a more unusual or complicated way. For example, we have a point $A=(1,2,3)$ and a straight line with this parametric equation:

$$x = t+1; y = 2t; z = t-1;$$

Is it possible to find the shortest distance from $A$ to the line by considering a point $H=(t+1,2t,t-1)$, the center of the sphere and $AH$ the radius of the sphere? Or find the symmetry of $A$ with respect to the line and find their midpoint $H$?

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With the sphere, we would have to calculate the minimum radius. This would be a tedious process. As far as the other "mirroring method" we would heve to do the computation below anyway because we would need to find a line through $A$ and perpendicular to our line given.

The simplest possibility is:

Take the direction vector of the line, $$\vec d= \vec i+2\vec j+ \vec k.$$ And take the vector pointing from a point of the line (belonging to $t$) to $A$. This vector is

$$\vec p_t = t\vec i +(2t-2)\vec j+(t-4)\vec k.$$

We are looking for a $t$ for which $\vec d$ and $\vec p_t$ are perpendicular. If they are perpendicular then their scalar product is zero:

$$(\vec d\cdot \vec p_t)=t+2(2t-2)+t-4=6t-8=0.$$

From here

$$t=\frac43$$

which is the parameter belonging to the closest point.

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    Thanks for the reply, i am aware of that method. The point of my question is to see if it is possible to calculate the minimum distance in different ways, (i am aware that there are 6 ways till now with 2 of them unknown to me) such as the ones i stated. – 2017-02-18