Scaling Vectors in Problems: What, When & Why?

[mill]

I've made mistakes where scaling was used but I just assumed that I didn't need it. e.g. a bug walking towards <1,1,1> is scaled to <1/sqrt(3), etc>. Under what kind of conditions/in what kind of problems should vectors be scaled? I know that v/|v| is the unit vector but how do I relate this to problems?

Furthermore, for this problem

A unit vector that is perpendicular to both v = <1; 3; 2> and w = <4; 2; 1> is...

I thought would be vxw, but the answer is (1/(5sqrt(6))vxw. Why was scaling needed here?

 

[SteamKing]

"A unit vector" means a vector whose magnitude is equal to 1.

Does the vector v x w from your example have a magnitude equal to 1?

 

[mill]

"A unit vector" means a vector whose magnitude is equal to 1.

Does the vector v x w from your example have a magnitude equal to 1?

No. Does this mean that I would need to scale everything not equal to 1? How is simply v x w different from the scaled v x w in finding the vector perpendicular to both?

 

[Matterwave]

You are asked to find a unit vector. When you are asked to find a unit vector, you have to scale your vector to have magnitude 1. If you are not asked to find a unit vector, you can leave the vector as is. The difference in vectors is just scaling...

 

[mill]

You are asked to find a unit vector. When you are asked to find a unit vector, you have to scale your vector to have magnitude 1. If you are not asked to find a unit vector, you can leave the vector as is. The difference in vectors is just scaling...

I see. But for that problem, even if unscaled, wouldn't v x w still be perpendicular to v and w? I guess I don't see how the scaled answer is the only correct one.

 

[Matterwave]

I see. But for that problem, even if unscaled, wouldn't v x w still be perpendicular to v and w? I guess I don't see how the scaled answer is the only correct one.

Yes, v x w is always perpendicular to both v and w. You are asked SPECIFICALLY to find the UNIT vector that is perpendicular to both v and w. The only one (actually 2 since there's one going the other way) is the scaled version. Unscaled versions are not UNIT vectors, but they are still perpendicular.

 

[mill]

Yes, v x w is always perpendicular to both v and w. You are asked SPECIFICALLY to find the UNIT vector that is perpendicular to both v and w. The only one (actually 2 since there's one going the other way) is the scaled version. Unscaled versions are not UNIT vectors, but they are still perpendicular.

I see.

How do scaled vectors function differently from simply vectors in problems (geometrically)? In what type of problems must scaling take place? Typically, distance problems? Other than the bug walking problem I've only seen it used with the helicopter flying in a certain direction. Do vectors need to be scaled in cases of finding distance away from a curve?

 

[Matterwave]

The scaled vectors are just shorter or longer than the unscaled vectors...there's no other difference.

It's good to use unit vectors because then your distances will come out in good units. This is like using a meter stick that is exactly 1 meter long. You are certainly free to use a meter stick that is 1.43 meters long, but all your measurements come out in multiples of 1.43 meters...

 

[mill]

The scaled vectors are just shorter or longer than the unscaled vectors...there's no other difference.

It's good to use unit vectors because then your distances will come out in good units. This is like using a meter stick that is exactly 1 meter long. You are certainly free to use a meter stick that is 1.43 meters long, but all your measurements come out in multiples of 1.43 meters...

Thanks. I think I got it. Just to clarify, it is still correct to use unscaled vectors in distance problems?

 

[Matterwave]

You can use whatever you want, as long as you make sure you know what you're doing, so you can give the correct answer in the end.

Just as the meter stick analogy. You are free to use a 1.43 meter long meter stick, as long as you know in the end to give your results in the correct way. So if something is 2 times as long as your 1.43 meter long meter stick, know that this thing is 2.86 meters long and NOT 2 meters long.

 

[mill]

Got it. Thanks.