Vector resolutes help

[Stick]

I had difficulty with parts 8c and 9c, but I worked them out in the end. I was wondering if it is a co-incidence that the shortest distance is also the magnitude of the perpendicular rectangular resolute, or if that's actually meant to happen. I drew out a diagram for these questions and it wasn't too helpful due to the whole 3D aspect and it actually hurt my brain. >_< Is there some sort of easier way to do this?

[Hancock]

Nope, your intuition is entirely correct. If you think about it, the smallest distance from a point to a line is going to be another line that is perfectly perpendicular to the original line in question from the point.



As we can see from the picture, the shortest distance is going to be from A to B. That is precisely the perpendicular component of the vector v with respect to u.

[Stick]

But what about this sort of triangle? The shortest distance from the top vertex to the bottom horizontal edge can't be a perpendicular vertical line (it is in fact the left edge). How do I tell when this is the case?

[Jenny_2108]

But what about this sort of triangle? The shortest distance from the top vertex to the bottom horizontal edge can't be a perpendicular vertical line (it is in fact the left edge). How do I tell when this is the case?

the shortest distance is still the vector resolute perpendicular (red line), doesn't matter right or left edge

[Stick]

Are you sure you can do that? The red line technically doesn't touch the bottom edge (you continued it on).

[Hancock]

Yes, but let's think about it such that the top vertex is the origin. Therefore, the shortest distance from point A (the bottom left vertex) will be along the line OA because that is a straight line to the intended destination. Sorry if this isn't helping much, geometry with vectors was never my strong suit.
 

[Stick]

So you're suggesting that I find the magnitude of the perpendicular rectangular resolute, but then check the magnitude of the edges? I guess that makes sense.

[Jenny_2108]

Are you sure you can do that? The red line technically doesn't touch the bottom edge (you continued it on).

From definition: vector resolute is in direction perpendicular and actually, you just extend the bottom line only. Because its a vector, we can extend it in the same direction

[Stick]

The question asks to find the shortest distance between the two points. I'll just remember that in most instances that the vector resolute is generally correct.