Final answer:
The distance between points A and B is the square root of 74. To convert Cartesian coordinates to polar coordinates, you calculate the magnitude r and the direction angle theta using the formulas r = sqrt(x^2 + y^2) and theta = atan2(y, x).
Step-by-step explanation:
Distance and Polar Coordinates between Two Points
To find the distance between two points in the Cartesian coordinate system, like points A(2.00 m, -4.00 m) and B(-3.00 m, 3.00 m), you apply the distance formula derived from Pythagoras’ theorem:
\(distance = \sqrt{ (x2 - x1)^2 + (y2 - y1)^2 }\)
Substituting the given coordinates:
\(distance = \sqrt{ (-3 - 2)^2 + (3 - (-4))^2 }\)
\(distance = \sqrt{ 25 + 49 }\)
\(distance = \sqrt{ 74 }\)
Next, to convert the Cartesian coordinates to polar coordinates, the following formulae are used:
\( r = \sqrt{ x^2 + y^2 }\)
\( \theta = atan2(y, x) \)
Where \(r\) is the magnitude and \(\theta\) is the direction angle in radians.
For point A:
\( r_A = \sqrt{ 2^2 + (-4)^2 }\)
\( r_A = \sqrt{ 20 }\)
\( \theta_A = atan2(-4, 2) \)
The angle \(\theta_A\) can be calculated using a calculator set to radian mode.