Lines and Planes

[maeveoneill]

Find the point R on l that is closest to Q. Q= (0,1,0), l with parametric equations x =1 -2t, y= 1, z= 1 + 3t.

I have figured out to do this for a point in a 2D. I just don't know how to write the equation of the line when its in 3D. If I could just get some help starting this off, that would be appreciated.

 

[royhaas]

Compute the (squared) distance from the line to the point. What is the minimum distance?

 

[pka]

The vector \(\displaystyle D = \left\langle { - 2,0,3} \right\rangle\) is the direction vector of the given line.
The point \(\displaystyle P = (1,1,1)\) is on the given line.
Define a new vector \(\displaystyle E = \overrightarrow {PQ} - \frac{{\overrightarrow {PQ} \cdot D}}{{D \cdot D}}D\).
Now find the point where the given line intersects the line \(\displaystyle Q + sE\).