What Is the Direction of Plane 1 Relative to Plane 2?

[bluefire90]

Two airplanes taxi as they approach the terminal. Plane 1 taxies with a speed of 11.3 m/s due north. Plane 2 taxies with a speed of 7.6 m/s in a direction 22.2° north of west. What is the direction of plane 1 relative to plane 2? Give the angle counterclockwise relative to the north. What is the direction of plane 2 relative to plane 1?

*I solved for magnitude first then velocity*

What is the magnitude of the velocity of plane 1 relative to plane 2?

To solve for the magnitude I found the vector components of plane 2:

Y component: 7.6 * sin(22.2°) = 2.87 m/s
X component: 7.6 * cos(22.2°)= 7.04 m/s
To find the final magnitude of the resultant vector I just used the pythagorean theorem after subtracting the plane 1 components FROM the plane 2 components.

2.87 m/s - 11.3 m/s = -8.43 m/s
7.04 m/s - 0 m/s = 7.04 m/s

Now to find the magnitude I used the pythagorean theorem.
Using the pythagorean theorom I got 10.98m/s as the relative velocity. This velocity is the same for plane 2 relative to plane 1.

To find the direction of plane 1 to plane 2 I just used the arctan formula

arctan (7.04/-8.43) = -39.86°
Now since its counterclockwise from north I just added to 360 to get 320° which is the correct answer.

I can't seem to figure out how to get the direction of plane 2 relative to plane 1 however and I've tried using the law of cosines and various other methods but I'm not sure where to go. Any help would be appreciated.

 

[tiny-tim]

Welcome to PF!

Hi bluefire90! Welcome to PF!

I can't seem to figure out how to get the direction of plane 2 relative to plane 1 however and I've tried using the law of cosines and various other methods but I'm not sure where to go. Any help would be appreciated.

Yes, your results look fine.

And the direction of plane 2 relative to plane 1 is just the opposite of the direction of plane 1 relative to plane 2 (V₂₁ = -V₁₂).

So the opposite of 320º is … ?

 

[bluefire90]

Thanks for the help! Although I did try -320.0° and it wasn't correct! I thought the same thing though

 

[tiny-tim]

Thanks for the help! Although I did try -320.0° and it wasn't correct! I thought the same thing though

There's no such thing as -320º … that would be 40º …

but that isn't the opposite of 320º anyway, is it?

(just draw it! )

 

[bluefire90]

Haha my background in geometry is pretty weak but if I looked at it logically and drew it the opposite might be 130?

 

[bluefire90]

140 degrees i mean

 

[tiny-tim]

Help! You're confusing me!

Yes, the opposite angle is always the original ± 180º.

 

[bluefire90]

haha thanks for the help! Wow that problem just made a lot more sense!