Two geological field teams are working in a remote area. A...

Two geological field teams are working in a remote area. A global positioning system (GPS) tracker at their base camp shows the location of the first team as $38 \mathrm{km}$ away, $19^{\circ}$ north of west, and the second team as 29 km away, $35^{\circ}$ east of north. When the first team uses its GPS to check the position of the second team, what does the GPS give for the second team's (a) distance from them and (b) direction, measured from due east?

Key Concepts

Polar Coordinates

Polar coordinates are used to represent the position of a point in a plane by specifying a distance from a reference point (the origin) and an angle from a reference direction. This concept is especially useful in navigation and GPS applications where locations are often given in terms of distance and bearing.

Cartesian Coordinates

Cartesian coordinates provide an alternative method for representing locations in a plane using two perpendicular axes, typically labeled x (horizontal) and y (vertical). Converting from polar to Cartesian coordinates involves using trigonometric functions, enabling easier computation of distances and relative directional differences between points.

Vector Subtraction

Vector subtraction is the process of determining the relative position or displacement between two points by subtracting one position vector from another. In the context of navigation, this technique is used to find the difference in location between two objects or teams from a common point of view.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are essential for converting between polar and Cartesian coordinates, as well as for calculating distances and angles. By applying these functions, one can determine the components of a vector along the horizontal and vertical axes and subsequently use these components to compute resultant distances and directions.

Transcript

00:01 All right, welcome back for this problem.

00:05 We have, we'll start with our coordinate system right here.

00:13 All right, now we have two research teams.

00:17 One is off in this direction.

00:21 Oops, that was supposed to start at the origin.

00:23 All right.

00:25 We'll have 19 degrees north of east, at a magnitude of 38 kilometers from the base here at the origin.

00:34 The second team magnitude of 29 kilometers an angle of 35 degrees east of north.

00:49 Alright so so far if you've been going through this chapter you've noticed that all of the angles that we're measuring here are formed along in relation to the x -axis so this one is a little funky right here and i forgot to mention we're trying to find make this green actually this result in the vector right here.

01:18 What is the distance between these two research teams? so as i was saying, this angle is formed with the y -axis and that's going to kind of change our math a little bit if we're using that signs and cosines.

01:38 The first thing i would do looking at this problem is i would just find this angle right here and then work with that angle.

01:44 So if you do 90 minus 35 degrees, you're going to get 55 degrees.

01:50 And then problem solved.

01:52 We can just work without angle instead.

01:55 All right.

01:56 So let's get working here.

02:00 So we've got the x component of these vectors, x component.

02:12 For the spectra on the left right here, it has magnitude of 38 kilometers times the cosine of 19.

02:24 38 times the cosine of 19, that is going to be equal to 36 .9, excuse me, 35 .9 kilometers.

02:40 And then the x component of the vector on the right here, 29 times the cosine of 55, that angle with the relation to the x axis, which that is going to be equal to 16 .6 kilometers.

02:58 We're gonna do the same thing for the y components.

03:02 Those.

03:04 All right so 38 times the sign of 19 degrees that gives us 12 .4 kilometers...

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