You are the Captain of the USS Gauss and you have been flying in the direction $2 i + j + k$ for quite awhile. You are currently at the point $(6, 3, 3)$. Your helper monkey Mojo just discovered that there is a great pizza parlor on the next approaching planet that delivers to passing spaceships. The coordinates of the pizza parlor are $(8, 8, 8)$. To keep the pizza as piping hot and fresh as possible, they put the pizza into an insulated capsule and fire it in the direction of the closest approach of passing spacecraft.
(a) What is the parametric equation describing your flight path?
(b) How far will your pizza travel from the pizza parlor to the point of interception with the USS Gauss?
(c) What is the parametric equation describing the flight path of your pizza?
How to do (b) part here. Should we follow 'shortest distance between skew lines' approach?
Note that the line of motion of the spaceship and that of the pizza trajectory cannot be skew, they have to intersect at some point.
Once you've done a), you will have the line's equation and you can take any general point on it in terms of a variable ; let P be such a point $$P (2t+6,t+3,t+3)$$ Now a parallel vector from the point $(8,8,8)$ will be: $$(2t-2)i+(t-5)j+(t-5)k$$ As this is the line of shortest distance, it must be perpendicular to the original line
hence, $$(2t-2).2+(t-5).1+(t-5).1=0$$ solving, we get $t=\frac{7}{3}$, plugging it in, we get the point $$P(\frac{32}{3},\frac{16}{3},\frac{16}{3})$$ Now can you proceed?