Question

A map suggests that Atlanta is 730 miles in the direction of the north-east from Dallas. The same map shows that Chicago is 560 miles in the direction of$21.0^{°}$ the west from Atlanta. Figure P3.24 shows the locations of these three cities. Use this information to find the displacement from Dallas to Chicago.

Short answer

The displacement from Dallas to Chicagois $788\text{miles at}48°$Northeast of Dallas.

Step-by-step solution

Define Vector Components

In a two-dimensional coordinate system, the x-component andy-component are commonly considered to be the components of a vector. It can be written as _{$V=\left(vx,vy\right)$}with V denoting the vector.

$$\begin{gathered} v_x=V\cos \theta \\ {v}_y=V\sin \theta \end{gathered}$$

Find the displacement vector

To calculate the distance between Dallas and Chicago, we must first determine the magnitude of the displacement vector that connects them. The components of displacement vectors from Dallas to Atlanta and Atlanta to Chicago are added together.

The displacement in direction

$$\begin{gathered} d_x=\left(730\times \cos 5°\right)-\left(560\times \sin 21°\right) \\ =527\text{miles} \end{gathered}$$

The displacement in direction

$$\begin{gathered} d_y=\left(730\times \sin 5°\right)+\left(560\times \cos 21°\right) \\ =586\text{miles} \end{gathered}$$

Displacement (overall)

_{$$\begin{gathered} d=\sqrt{{527}^2+{586}^2} \\ =788\text{miles} \end{gathered}$$}