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.

To solve this problem, you should break down the velocities into their respective north-south and east-west components.

First, let's consider the plane's attempted velocity of 500 km/h[N]. In the north-south direction, this is simply 500 km/h since it is directly north. In the east-west direction, there is no velocity component, so it is 0 km/h.

Next, let's consider the true velocity of the plane, which is given as 450 km/h[N35°E]. To find the north-south component, you need to find the component of the velocity in the north direction. This can be calculated using trigonometry: north-south component = 450 km/h * sin(35°). Evaluating this gives you approximately 258.1 km/h[N].

To find the east-west component, you need to find the component of the velocity in the east direction. Again, using trigonometry: east-west component = 450 km/h * cos(35°). Evaluating this gives you approximately 366.2 km/h[E].

Now, let's consider the velocity of the wind. Since the wind is opposing the plane's motion, we can subtract the plane's true velocity from its attempted velocity to determine the wind's effect.

For the north-south component, the wind's effect is the difference between the two north-south components: wind's north-south component = 500 km/h - 258.1 km/h ≈ 241.9 km/h[S].

For the east-west component, the wind's effect is the difference between the two east-west components: wind's east-west component = 0 km/h - 366.2 km/h ≈ -366.2 km/h[W].

Therefore, the velocity of the wind is approximately 241.9 km/h[S] and -366.2 km/h[W].