Question

Suppose you first walk 12.0 m in a direction 20? west of north and then 20.0 m in a direction 40.0? south of west. how far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position?

Answer 1

The representation of this problem is shown in Figure 1. So our goal is to find the vector $\overrightarrow{R}$. From the figure we know that:

$\left | \overrightarrow{A} \right |=12m \\ \\ \left | \overrightarrow{B} \right |=20m \\ \\ \theta_{A}=20^{\circ} \\ \\ \theta_{B}=40^{\circ}$

From geometry, we know that:

$\overrightarrow{R}=\overrightarrow{A}+\overrightarrow{B}$

Then using vector decomposition into components:

$For \ A: \\ \\ A_x=-\left | \overrightarrow{A} \right |sin\theta_A=-12sin(20^{\circ})=-4.10 \\ \\ A_y=\left | \overrightarrow{A} \right |cos\theta_A=12cos(20^{\circ})=11.27 \\ \\ \\ For \ B: \\ \\ B_x=-\left | \overrightarrow{B} \right |cos\theta_B=-20cos(40^{\circ})=-15.32 \\ \\ B_y=-\left | \overrightarrow{B} \right |sin\theta_B=-20sin(40^{\circ})=-12.85$

Therefore:

$R_x=A_x+B_x=-4.10-15.32=-19.42m \\ \\ R_y=A_y+B_y=11.27-12.85=-1.58m$

So if you want to find out how far are you from your starting point you need to know the magnitude of the vector $\overrightarrow{R}$, that is:

$\left | \overrightarrow{R} \right |= \sqrt{R_x^2+R_y^2}=\sqrt{(-19.42)^2+(-1.58)^2}=\boxed{19.48m}$

Finally, let's find the compass direction of a line connecting your starting point to your final position. What we are looking for here is an angle that is shown in Figure 2 which is an angle defined with respect to the positive x-axis. Therefore:

$\theta_R=180^{\circ}+tan^{-1}(\frac{\left | R_y \right |}{\left | R_x \right |}) \\ \\ \theta_R=180^{\circ}+tan^{-1}(\frac{1.58}{19.42}) \\ \\ \theta_R=180^{\circ}+4.65^{\circ}=185.85^{\circ}$

Answer 2

Representation of the given information in vector form can be used to find

the distance and direction from the start to the final position.

• The distance from the starting point is approximately 19.5 meters

• The compass direction of the line connecting the starting point and final position is 4.65° South of West

Reasons:

Distance, w₁, walked in the direction 20° West of North = 12.0 m

Distance, w₂, walked in the direction 40.0° South of West = 20.0 m

Distance and direction from the starting location; Required

Solution:

The distances presented in vector form are;

$\overrightarrow{w_1}$ = -12.0×sin(20°)·i + 12.0×cos(20°)·j

$\overrightarrow{w_2}$ = -20.0×cos(40°)·i - 20.0×sin(40°)·j

Therefore, we have;

R = $\overrightarrow{w_1}$ + $\overrightarrow{w_2}$ = (-12×sin(20°) + -20×cos(40°))·i + (12×cos(20°) + - 20×sin(40°))·j

R = -19.43·i - 1.58·j

Therefore;

|R| = √((-19.43)² + (-1.58)²) ≈ 19.5

The distance from the starting point is |R| ≈ 19.5 m

The tangent of the angle θ formed by the line connecting your starting

point to your final position is given as follows;

$tan(\theta) = \dfrac{Opposite \ side \ length\ to \ \angle\theta}{Adjacent\ side \ length\ to \ \angle\theta}$

The opposite side length = The distance moved along the positive y-

direction = The j coefficient

The adjacent side length = The distance moved along the positive x-direction = The i coefficient

Therefore;

$tan(\theta) = \left(\dfrac{-1.58}{-19.43} \right)$

The direction is therefore;

$\theta = arctan \left(\dfrac{-1.58}{-19.43} \right) \approx 4.65^{\circ}$

Therefore, the compass direction of a line connecting the starting and final

position is 4.65° South of West.

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