Question

a postal employee drives a delivery truck along the route shown in (figure 1). for related problem - solving tips and strategies, you may want to view a video tutor solution of adding two vectors at right angles. part a determine the magnitude of the resultant displacement by drawing a scale diagram. express your answer in kilometers. part b determine the direction of the resultant displacement. express your answer in degrees. north of east

Explanation:

Step1: Resolve vectors into components

Let the first - displacement vector $\vec{A}$ be $2.6\ km$ east, so $\vec{A}=(2.6,0)\ km$. The second - displacement vector $\vec{B}$ is $4.0\ km$ north, so $\vec{B}=(0,4.0)\ km$. The third - displacement vector $\vec{C}$ has magnitude $3.1\ km$ at an angle of $45^{\circ}$ south - east. Its $x$ - component is $C_x = 3.1\cos45^{\circ}\ km\approx3.1\times\frac{\sqrt{2}}{2}\ km\approx2.19\ km$ and its $y$ - component is $C_y=- 3.1\sin45^{\circ}\ km\approx - 2.19\ km$.

Step2: Calculate the total $x$ - component of the resultant vector

$R_x=A_x + B_x+C_x=2.6 + 0+2.19 = 4.79\ km$

Step3: Calculate the total $y$ - component of the resultant vector

$R_y=A_y + B_y + C_y=0 + 4.0-2.19 = 1.81\ km$

Step4: Calculate the magnitude of the resultant vector

Use the Pythagorean theorem $R=\sqrt{R_x^{2}+R_y^{2}}=\sqrt{(4.79)^{2}+(1.81)^{2}}=\sqrt{22.9441 + 3.2761}=\sqrt{26.2202}\approx5.12\ km$

Step5: Calculate the direction of the resultant vector

Use the formula $\theta=\tan^{-1}(\frac{R_y}{R_x})=\tan^{-1}(\frac{1.81}{4.79})\approx20.6^{\circ}$ north of east

Answer:

Part A: $5.12\ km$
Part B: $20.6^{\circ}$ north of east