Question

A pizza delivery driver must make three stops on her route. She will first leave the restaurant and travel 4 km due north to the first house. The next house is 6 km away at 45° south of west according to her map. The final stop is 5 km away at 60° north of west. What is her displacement from the restaurant to the final stop?

Answer 1

5.3 km

Explanation:

       This question involves continuous displacement in various directions. When it becomes difficult to imagine, vector analysis becomes handy.

       Let us denote each of the individual displacements by a vector. Consider the unit vectors $\vec{i}\textrm{ and }\vec{j}$ as the unit vectors in the direction of East and North respectively.

       By simple calculations, we can derive the unit vectors $\vec{j},\frac{-\vec{i}-\vec{j}}{2}\textrm{ and }\frac{-\frac{1}{2}\vec{i}+\frac{\sqrt{3}}{2}\vec{j}}{2}$ in the directions North, $45^{o}$ South of West and $60^{o}$ North of West respectively.

       So Total displacement vector = Sum of individual displacement vectors.

       Displacement vector = $4(\vec{j})+6(\frac{-\vec{i}-\vec{j}}{2})+5(\frac{-\frac{1}{2}\vec{i}+\frac{\sqrt{3}}{2}\vec{j}}{2})=-4.25\vec{i}+3.165\vec{j}$

       Magnitude of Displacement = $|-4.25\vec{i}+3.165\vec{j}|=5.3km$

∴ Total displacement = $5.3km$