Question

Solve the following problem using analytical techniques: Suppose you walk $\mathbf{18}\mathbf{.}\mathbf{0}\mathit{m}$straight west and then $\mathbf{250}\mathbf{.}\mathbf{0}\mathit{m}$straight north. 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? (If you represent the two legs of the walk as vector displacements and , as in Figure, then this problem asks you to find their sum $\mathit{R}\mathbf{=}\mathit{A}\mathbf{+}\mathit{B}$.)

The two displacements and add to give a total displacement having magnitude and direction .

Note that you can also solve this graphically. Discuss why the analytical technique for solving this problem is potentially more accurate than the graphical technique.

Short answer

The distance from the starting point is $30.8m$, and the compass direction of a line connecting starting point to the final position is $54.2°$north of west.

Step-by-step solution

Definition of vector

Vectors, which have both magnitude and direction, are physical quantities. Two vectors cannot be added using simple algebraic rules; they can be added by using the triangle law of vector addition.

Given data:

• A =$18.0m$
• B = .$25.0m$

Magnitude of resultant vector

The magnitude of the resultant vector is

$R=\sqrt{A^2+B^2}$

Here is $A$the magnitude of the vector$A$ (displacement towards west) and$B$ is the magnitude of the vector $B$(displacement towards north).

Substituterole="math" localid="1668686725834" $18.0m$ for$A$ and$25.0m$ for $B$.

$$\begin{gathered} R=\sqrt{{(18.0m)}^2+{(25.0m)}^2} \\ =30.8m \end{gathered}$$

Direction of compass

The direction of the compass is

$\theta ={\tan }^{-1}\left(\frac{B}{A}\right)$

Substitute$18.0m$ for $A$and $25.0m$for$B$ .

$$\begin{gathered} \theta ={\tan }^{-1}\left(\frac{25.0m}{18.0m}\right) \\ =54.2° \end{gathered}$$

Hence, the distance from the starting point is 30.8m and the compass direction of a line connecting starting point to the final position is $54.2°$north of west.

Reason for using analytical technique

The analytical methods are more concise, accurate, and precise than the graphical method, which is limited by the accuracy with which a drawing can be made.

Hence, the analytical methods are used because of more precision.