HW 6, Questions 7-8

I’m having trouble figuring out the last 2 questions on the homework, has anyone had any luck with it?

#7

#8

For #7, I broke \vec {r_1}(t) down into its parametric form and solved those as a two-variable system of equations. So 10+4t=-12.5-3s and 7+2t=4.5+s, etc. Once you solve for s, you can plug that back into t and solve. Once you have a numerical value for t, you can plug that into \vec {r_2}(t) to find P.

For #8, you can first find the vector \overrightarrow{QP} where Q is a point on the line and then apply this formula:

h = \frac{|| {\overrightarrow{QP}\times \vec d}||}{{||d||}}

@aaudia I made a little edit to your post so that it matches the rest of what you said about problem #7. If I understand correctly, you want to solve

for s and t and then see if that works in the third equation. Effectively, you’re solving the system

You don’t want to solve \vec{r}_1(t) = \vec{r}_2(t). That checks if the points collide but the paths might intersect without the points being at the same place at the same time.

For #8, my expectation was that you would form a vector \vec{v} from some point on the line (L(0), for example) to the point P and that you would the magnitude of \vec{v}_\perp, where

decomposes v into a vector parallel to the line and a vector perpendicular to the line.

@mark I think I see what you’re saying. The confusion then came from me using the free variable t in \vec r_{2}(t) as it appears in the problem instead of \vec r_{2}(s) as would be consistent with our system. Would that make sense?