A sailboat sets out from the U.S. side of Lake Erie for a point on the Canadian side,90.0 km due north. The sailor, however, ends up 50.0 kmdue east of the starting point. (a) How far and (b) in what direction must the sailor now sail to reach the original destination?

The point original destination point is 90 km north of the starting point.

The present location point is 50 km towards the East of the starting.

Two vectors can be added using the law of vector addition. Vector diagrams are used to represent the vectors and find the addition or subtraction of two or more vectors.

Using a vector diagram and the Pythagorean Theorem we can find the distance between two points and the angle between them.

Formulae:

The Pythagorean Theorem,

$$\begin{gathered} c=\sqrt{a^2+b^2} \\ \tan \theta =\frac{b}{a} \end{gathered}$$

The distance of the sailor from the original destination is,

$$\begin{gathered} c=\sqrt{a^2+b^2} \\ =\sqrt{{90}^2+{50}^2} \\ =103\mathrm{km} \end{gathered}$$

The direction of the original destination point from the present location of the sailor is given by the angle $\theta$.

From the Pythagorean Theorem,

$$\begin{gathered} \tan \theta =\frac{b}{a} \\ \theta ={\tan }^{-1}\left(\frac{50}{90}\right) \\ ={\tan }^{-1}\left(0.55\right) \\ =29.1° \end{gathered}$$

So, the direction is$29.1°$ west of north.