A plane tries to head away from an airport at 500km/h[N], but finds that its true velocity is 450km/h[N35degreesE] due to headwinds. What is the velocity of the wind?

I'm supposed to be using components, so for N-S I used 500+450sin(55 which got me 868.6 then, for E-W I did 450cos(55 and that was 258.1. I then used pythagorean theorem and square rooted 868.6^2+258.1^2 and that was 906 km/h.
Then, I did the inverse tan of 25831/868.6 and got 16.5 as the angle. Howeverm this would mean that the wind's velocity is 906 km/h[N16.5degreesE], but this is a headwind, and I think it should be coming in an oppoisng direction, right? I'm just really confused and don't really know what to do. Please, please help me understand this.

It is not clear from the question what the "true" velocity means, and what the direction of the runway is. If it is the air speed, then there must be a tail wind, not a head wind. This is a vector addition question, but I hesitate to try to solve it because I might be solving the wrong problem.

oh, okay, well, thanks for trying; I wish I could givr you more information, but that's all my worksheet says...

I can help you understand this problem. Let's break it down step by step.

First, we need to understand the concept of vector addition. In this problem, we have two vectors: the velocity of the plane and the velocity of the wind. The resulting velocity (true velocity) of the plane is the vector sum of these two velocities.

To find the velocity of the wind, we need to subtract the velocity of the plane from the true velocity.

Now let's calculate it step by step using components.

1. Start by breaking down the velocities into their North-South (N-S) and East-West (E-W) components.

Velocity of the plane:
N-S component = 500 km/h
E-W component = 0 km/h (since it is heading directly away from the airport)

True velocity of the plane:
Magnitude = 450 km/h
Direction = N35°E

To calculate the N-S and E-W components of the true velocity, we need to use trigonometry.

N-S component = 450 km/h * sin(35°) ≈ 450 km/h * 0.5736 ≈ 258.12 km/h
E-W component = 450 km/h * cos(35°) ≈ 450 km/h * 0.8192 ≈ 368.64 km/h

2. Now we can find the N-S and E-W components of the velocity of the wind by subtracting the components of the plane's velocity from the components of the true velocity.

N-S component of the wind = N-S component of true velocity - N-S component of the plane's velocity
= 258.12 km/h - 500 km/h
≈ -241.88 km/h

E-W component of the wind = E-W component of true velocity - E-W component of the plane's velocity
= 368.64 km/h - 0 km/h
= 368.64 km/h

3. Finally, we can calculate the magnitude and direction of the wind's velocity using the Pythagorean theorem and trigonometry.

Magnitude of the wind's velocity = √((-241.88 km/h)^2 + (368.64 km/h)^2) ≈ 439.53 km/h

Direction of the wind = arctan(E-W component of the wind / N-S component of the wind)
= arctan(368.64 km/h / -241.88 km/h)
≈ -58.19°

Note: The negative sign of the N-S component of the wind indicates that it is blowing in the opposite direction (from the south) of the plane's intended direction (north).

Therefore, the velocity of the wind is approximately 439.53 km/h [S58.19°E].