2.1 Vectors in the plane  (Page 7/29)

To find the effect of combining the two forces, add their representative vectors. First, express each vector in component form or in terms of the standard unit vectors. For this purpose, it is easiest if we align one of the vectors with the positive x -axis. The horizontal vector, then, has initial point $\left(0,0\right)$ and terminal point $\left(300,0\right).$ It can be expressed as $⟨300,0⟩$ or $300\text{i}.$

The second vector has magnitude $150$ and makes an angle of $15\text{°}$ with the first, so we can express it as $⟨150\text{cos}\left(15\text{°}\right),150\text{sin}\left(15\text{°}\right)⟩,$ or $150\text{cos}\left(15\text{°}\right)\text{i}+150\text{sin}\left(15\text{°}\right)\text{j}.$ Then, the sum of the vectors, or resultant vector, is $\text{r}=⟨300,0⟩+⟨150\text{cos}\left(15\text{°}\right),150\text{sin}\left(15\text{°}\right)⟩,$ and we have

The angle $\theta$ made by $\text{r}$ and the positive x -axis has $\text{tan}\theta =\frac{150\text{sin}15\text{°}}{\left(300+150\text{cos}15\text{°}\right)}\approx 0.09,$ so $\theta \approx ta{n}^{\mathrm{-1}}\left(0.09\right)\approx 5\text{°},$ which means the resultant force $\text{r}$ has an angle of $5\text{°}$ above the horizontal axis.

Let’s start by sketching the situation described ( [link] ).

Set up a sketch so that the initial points of the vectors lie at the origin. Then, the plane’s velocity vector is $\text{p}=\mathrm{-425}\text{i}.$ The vector describing the wind makes an angle of $225\text{°}$ with the positive x -axis:

When the airspeed and the wind act together on the plane, we can add their vectors to find the resultant force:

The magnitude of the resultant vector shows the effect of the wind on the ground speed of the airplane:

As a result of the wind, the plane is traveling at approximately $454$ mph relative to the ground.

To determine the bearing of the airplane, we want to find the direction of the vector $p+w\text{:}$

The overall direction of the plane is $3.57\text{°}$ south of west.