Question

You are camping with two friends, Joe and Karl. Since all three of you like your privacy, you don't pitch your tents close together. Joe's tent is 19.0 m from yours, in the direction 19.0° north of east. Karl's tent is 45.0 m from yours, in the direction 39.0° south of east. What is the distance between Karl's tent and Joe's tent?

Answer 1

Answer:

Distance between Karl and Joe is 38.467 m

Solution:

Let us assume that you are at origin

Now, as per the question:

Joe's tent is 19 m away from yours in the direction $19.0^{\circ}$ north of east.

Now,

Using vector notation for Joe's location, we get:

$\vec{r_{J}} = 19cos(19.0^{\circ})\hat{i} + 19sin(19.0^{\circ})\hat{j}$

$\vec{r_{J}} = 17.96\hat{i} + 6.185\hat{j} m$

Now,

Karl's tent is 45 m away from yours and is in the direction $39.0^{\circ}$south of east, i.e.,  $- 39.0^{\circ}$ from the positive x-axis:

Again,  using vector notation for Karl's location, we get:

$\vec{r_{K}} = 45cos(-319.0^{\circ})\hat{i} + 45sin(- 39.0^{\circ})\hat{j}$

$\vec{r_{K}} = 34.97\hat{i} - 28.32\hat{j} m$

Now,  obtain the vector difference between $\vec{r_{J}}$ and $\vec{r_{K}}$:

$\vec{r_{K}} - \vec{r_{J}} = 34.97\hat{i} - 28.32\hat{j} - (17.96\hat{i} + 6.185\hat{j}) m$

$\vec{d} = \vec{r_{K}} - \vec{r_{J}} = 17.01\hat{i} - 34.51\hat{j} m$

Now, the distance between Karl and Joe, d:

|\vec{d}| = |17.01\hat{i} - 34.51\hat{j}|

$d = \sqrt{(17.01)^{2} + (34.51)^{2}} m$

d = 38.469 m

The distance between Karl's and Joe's tent is: