Ruth sets out to visit her friend Ward, who lives 50 mi north and 100 east of her. She starts by driving east but after 30 mi she comes to a detour that takes her 15 I south before going east again. She then drives east for 8 mi and runs out of gas, so Ward flies there in a small plane to get her. What is Wards displacement vector? Give your answer a() in component form, using a coordinate system in which the y axis points north and (b) as a magnitude and direction.

The above equation is written because from the given data, she lives 100miles east of her which is on the positive x-axis and 50 miles north which is on the positive y-axis.

At Ruth's home, the position vector is given as

[tex]x_2=38i-15j[/tex]

The above equation is written because she is moving closer to the positive x-axis in 30 miles after which she also flew east in 8 miles. Also, she went south in 15 miles which is down the negative y-axis.

The Ward's displacement vector is calculated as

[tex]\begin{gathered} x=x_2-x_1 \\ =(38i-15j)-(100i+50j) \\ =-62i-65j \end{gathered}[/tex]

In component form, it can be written as

[tex]x=(-62,-65)[/tex]

(b)

The magnitude of Ward's displacement vector is calculated as

[tex]\begin{gathered} \lvert x\rvert=\sqrt[]{(-62)^2+(-65)^2} \\ =89.82\text{ m} \\ \approx90\text{ m} \end{gathered}[/tex]

Hence, the magnitude of Ward's displacement vector is 90 m

The direction of Ward's displacement vector is calculated as

[tex]\begin{gathered} \theta=\tan ^{-1}(\frac{-62}{-65}) \\ =43.64^0 \end{gathered}[/tex]