Extremely confusing vector addition problem

[mikky05v]

Homework Statement

Suppose a plane flies at a constant groundspeed of 500 miles per hour due east and encounters a 50 mile-per-hour wind from the northwest. Both the airspeed and the compass direction must change to for the plane to maintain its groundspeed and eastward direction. Find the airspeed to maintain its groundspeed and eastward direction. Round your answer to two decimal places.

Homework Equations

A vector U=<x,y>
||U|| means the length of U = √(x^2 +y^2). vectors can also be written as U = ||U||(cosθi, sinθj)

The Attempt at a Solution

I'm really not sure where to start. i think it is going to end up being similar to https://www.physicsforums.com/showthread.php?t=709519 my other problem question but I can't figure out what to do with this one.

I tried naming the vectors V is the new direction of the plane, V1 is the original direction along the X axis to the right, V2 is the wind direction which is at a 45 angle in the 4th quad because it is coming from the north west.

Going with this logic I did V=V1+V2
V1= 500(cos0i+sin0j)
V2= 50(cos315i+sin315j)

distribute the speeds, combine like terms and factor out i and j
V= <(500cos0+50cos315)i + (500sin0+50sin315)j> = <535.36i + (-35.36)j>

Then I plugged them into the distance forumla to find ||V|| which should be my speed and got
√535.36^2 + (-35.36)^2 = 536.53 which isn't an option.

I think my issue could be in the angles I'm using as that was kind of an assumption or maybe it's in my comprehension of air speed and ground speed.

second attempt
ok so I tried to use the V_{over ground} = V_{through the medium} + V_{wind or current} to find my answer

I shortened them to Vo=Vt+Vw
Vt=Vo-Vw
Vo = 500(cos0i+sin0i)
Vt = ||Vt||(cos∂i+sin∂j)
Vw = 50(cos315i+sin315j)

Vt = < (500cos0-50cos315)i + (500sin0-50sin315)j > = <429.29i +35.36j>
||Vt|| = sqrt ( 429.29^2 +35.36^2) = 430.74

That's not one of the possible answers. Are my angles wrong or am i missing something else?

 

[vela]

Homework Statement

Suppose a plane flies at a constant groundspeed of 500 miles per hour due east and encounters a 50 mile-per-hour wind from the northwest. Both the airspeed and the compass direction must change to for the plane to maintain its groundspeed and eastward direction. Find the airspeed to maintain its groundspeed and eastward direction. Round your answer to two decimal places.

Homework Equations

A vector U=<x,y>
||U|| means the length of U = √(x^2 +y^2). vectors can also be written as U = ||U||(cosθi, sinθj)

The Attempt at a Solution

I'm really not sure where to start. i think it is going to end up being similar to https://www.physicsforums.com/showthread.php?t=709519 my other problem question but I can't figure out what to do with this one.

I tried naming the vectors V is the new direction of the plane, V1 is the original direction along the X axis to the right, V2 is the wind direction which is at a 45 angle in the 4th quad because it is coming from the north west.

Going with this logic I did V=V1+V2
V1= 500(cos0i+sin0j)
V2= 50(cos315i+sin315j)

distribute the speeds, combine like terms and factor out i and j
V= <(500cos0+50cos315)i + (500sin0+50sin315)j> = <535.36i + (-35.36)j>

Then I plugged them into the distance forumla to find ||V|| which should be my speed and got
√535.36^2 + (-35.36)^2 = 536.53 which isn't an option.

I think my issue could be in the angles I'm using as that was kind of an assumption or maybe it's in my comprehension of air speed and ground speed.

second attempt
ok so I tried to use the V_{over ground} = V_{through the medium} + V_{wind or current} to find my answer

I shortened them to Vo=Vt+Vw
Vt=Vo-Vw
Vo = 500(cos0i+sin0i)
Vt = ||Vt||(cos∂i+sin∂j)
Vw = 50(cos315i+sin315j)

Vt = < (500cos0-50cos315)i + (500sin0-50sin315)j > = <429.29i +35.36j>
||Vt|| = sqrt ( 429.29^2 +35.36^2) = 430.74

That's not one of the possible answers. Are my angles wrong or am i missing something else?

You somehow got the wrong number for the x-component of Vt. It comes out to 464 mph.