Another Vector Problem

A 100-meter dash is run on a track in the drection of the vector v = 2i + 6j. The wind velocity w is a vector w = 5i + j km/hr. The rules say that a legal wind speed measured in the direction of ther dash must not exceed 5km/hr. Will the race results be disqualified due to an illegal wind speed?

I know how to approach this problem I need to find the magintude of the vector component of the wind that is perpendicular to the race track.

I'm a little confused about the application of the formula, can I do v * (w/||w||), and if so, can you please explain this.

Brian
Math / Science

Last Comment
doraiswamy

8/22/2022 - Mon
BrianGEFF719

What is the difference between:

v * (w/||w||) and w * (v/||v||) ??
ozo

The they differ by a factor of ||w||/||v||
ozo

I think you want v/||v|| because the length of v does not matter, while the length of w does.
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rwheeler23
BrianGEFF719

>>I think you want v/||v|| because the length of v does not matter, while the length of w does.

Thats an interesting point, I didnt think about that.
ozo

Although if you go into the physics, cross wind also affects runners by increasing their drag.
doraiswamy

Brian=>"I need to find the magintude of the vector component of the wind that is perpendicular to the race track."

Don't you need to find the magnitude  of the vector component of the wind parallel to the race track?

Basically, the DOT Product of a vector X with a unit vector Y gives the component of X in the direction of Y. In this case, the unit vector in the direction of the race track is V/|V|.

Therefore you need W.(V/|V|)

HTH
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BrianGEFF719

Ok, it has been a few days since i've been to this problem, I just have a quick question and then I will finalize.

v/||v|| is simply a unit vector that points in the direction of the race track...

Why will dotting a unit vector that points in the direction of the race with the wind vector give me the magnitude of the component of the wind vector parrallel to the race?

Brian
ozo

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doraiswamy

Brian,

The dot product of 2 vectors X and Y is a scalar with a magnitude of |X||Y|cos(theta) where theta is the angle between the two vectors. Now, if Y is the unit vector, then |Y| is 1. In this case, the product will be |X|cos(theta), which is the component of X in the direction of Y.

In your case, we want to find out the component of the wind which aids or retards running along the race track. Thus we are looking for the component which is along i.e. parallel to the race track. This is given by the dot product of the wind vector and the unit vector along the direction of the ract track, and that is V/|V|. Take the dot product of this with the vector W, and there you have it.