An ant is crawling along a yardstick that is pointed with the 0-inch mark to the east and the 36-inch mark to the west. It starts at the 16 inch mark, crawls to the 29-inch mark, then moves to the 14-inch mark. What is the total distance the ant traveled? What was the total displacement of the ant?

Lets put a frame of reference in the problem. Starting the frame of reference at the point with the 0-inch mark, and making the unit vector [tex]\hat{i}[/tex] pointing in the west direction, the ant start at position

[tex]\vec{r}_0 = 16 \ inch \ \hat{i}[/tex]

Then, moves to

[tex]\vec{r}_1 = 29 \ inch \ \hat{i}[/tex]

so, the distance traveled here is

[tex]d_1 = |\vec{r}_1 - \vec{r}_0 | = | 29 \ inch \ \hat{i} - 16 \ inch \ \hat{i} |[/tex]

[tex]d_1 = | 13 \ inch \ \hat{i} |[/tex]

[tex]d_1 = 13 \ inch [/tex]

after this, the ant travels to

[tex]\vec{r}_2 = 14 \ inch \ \hat{i}[/tex]

so, the distance traveled here is

[tex]d_2 = |\vec{r}_2 - \vec{r}_1 | = | 14 \ inch \ \hat{i} - 29 \ inch \ \hat{i} |[/tex]

[tex]d_2 = | - 15 \ inch \ \hat{i} |[/tex]

[tex]d_2 = 15 \ inch [/tex]

The total distance traveled will be:

[tex]d_1 + d_2 = 13 \ inch + 15 \ inch = 28 \ inch[/tex]

The displacement is the final position vector minus the initial position vector:

[tex]\vec{D}=\vec{r}_2 - \vec{r}_1[/tex]

[tex]\vec{D}= 14 \ inch \ \hat{i} - 16 \ inch \ \hat{i}[/tex]

[tex]\vec{D}= - 2 \ inch \ \hat{i}[/tex]

This is 2 inches to the east.