Determine the vector equation of a line passing through the point P(3,2,-1) and with a direction vector perpendicular to the line r=(2,-3,4)+s(1,1,-2), seR

Answer 1

would the answer be r-(3,2,-1)+s(1,1,1)

Answer 2

Let's check:
r1=(3,2,-1)+s<1,1,1>
When s=0, r passes through the point (3,2,-1).
Now check the product of the two direction vectors:
<1,1,1> dot <1,1,-2>
=1+1-2
=0
So the two lines are perpendicular
and r1 satisfies the required conditions.

Note: multiple lines can be perpendicular r, for example:
r2=(3,2,-1)+t<5,3,4>
would also work, since
<5,3,4>dot<1,1,-2>=0 as well.