Solve Cartesian and Polar Coordinates, Distance, and Direction Problems"

[graffz]

1) Two points in the xy plane have Cartesian coordinates (1.00, -4.00) m and (-5.00, 5.00) m. Determine their polar coordinates @ (-5.00, 5.00)
r = ?
θ = ?

2) A plane flies from base camp to Lake A, 290 km away in the direction 20.0° north of east. After dropping off supplies it flies to Lake B, which is 230 km 30.0° west of north from Lake A. Graphically determine the distance and direction from Lake B to the base camp.
Distance = ?
Direction = ?

3) Given the vectors = 5.00 + 8.00 and = 3.00 - 2.00.
vector D
r =
θ =


I have absolutely no clue how to solve these, or where to begin. I don't have my book yet since it has not been delivered, and I don't mean to be a pest, but I really need help. Thank you!

 

[graffz]

AND ANOTHER ONE, PLEASE.

An air-traffic controller observes two aircraft on his radar screen. The first is at altitude 850 m, horizontal distance 19.8 km, and 25.5° south of west. The second aircraft is at altitude 1000 m, horizontal distance 18.0 km, and 19.0° south of west. What is the distance between the two aircraft? (Place the x-axis west, the y-axis south, and the z axis vertical.)

 

[nlightner]

I am happy to help you with these problems. Let's start by defining some terms and concepts that will be useful in solving these problems.

Cartesian coordinates are a way of representing points in a two-dimensional plane using x and y coordinates. The x-coordinate represents the horizontal distance from the origin, while the y-coordinate represents the vertical distance from the origin.

Polar coordinates, on the other hand, use a distance (r) and an angle (θ) to represent a point in a two-dimensional plane. The distance r is measured from the origin to the point, and the angle θ is measured from the positive x-axis in a counterclockwise direction.

Now, let's solve the first problem. We are given two points in the xy plane with Cartesian coordinates (1.00, -4.00) m and (-5.00, 5.00) m. To find their polar coordinates, we need to use the following formulas:

r = √(x^2 + y^2)
θ = tan^-1 (y/x)

Plugging in the values from the given points, we get:

r = √(1.00^2 + (-4.00)^2) = √(1 + 16) = √17 m
θ = tan^-1 (-4.00/1.00) = tan^-1 (-4) = -75.96°

Therefore, the polar coordinates of the point (-5.00, 5.00) m are (r, θ) = (√17 m, -75.96°).

Moving on to the second problem, we have a plane flying from base camp to Lake A, 290 km away in the direction 20.0° north of east. After dropping off supplies, it flies to Lake B, which is 230 km 30.0° west of north from Lake A. To find the distance and direction from Lake B to the base camp, we can use the following steps:

1. Draw a diagram representing the given information. Label the points A, B, and the base camp.

2. Use the law of cosines to find the distance from Lake B to the base camp. The law of cosines states that c^2 = a^2 + b^2 - 2ab cos(C), where c is the side opposite angle C. In this case, we want to find