Vectors: Finding line thats perpendicular to 2 lines

[H.Bisho18]

I'm stuck on 16.
I've written 2 vector equations but i'm just not sure how to get a line to pass through both of them perpendicularly
Here's my working so far but its not much, pretty stumped :/

thanks

 

[Dr.Peterson]

Write an expression for the vector from P to Q, as a function of your two variables [MATH]\lambda[/MATH] and [MATH]\mu[/MATH]. What must be true if this vector is perpendicular to the vectors you called a and b? You might think in terms of either the scalar product or the vector product.

 

[pka]

View attachment 13436
I'm stuck on 16. I've written 2 vector equations but i'm just not sure how to get a line to pass through both of them perpendicularly

This is the most confuted question. These are two skew lines.
\(\displaystyle \ell(s)=(1,1,2)+s<2,1,-1>~\:\&~\;\ell(t)=(1,1,4)+y<1,1,-2>\)
\(\displaystyle \ell(s):\:x=1+2s,\:y=1+s,\:z=2-s~\:\&~\;\ell(t):\:x=1+t\:y=1+t\:z=4-2t\)

to H.Bisho18, can you show that these two are skew lines?
Two skew lines have a unique common perpendicular.
That is the \(\displaystyle \overline{PQ}\) that you are asked to find.