An adventurous aardvark arduously ambles from (3.0, 2.0) m to (−4.0, −2.0) m. What is the displacement in polar coordinates?
change in x = -7
change in y =-4
d^2 = x^2+y^2 = 49+16 = 65
so r = sqrt 65
angle in quadrant 3
tan angle below -x axis = -4/-7
so angle below x axis = 29.7
theta = 180+29.7 = 209.7
To find the displacement in polar coordinates, we need to convert the given Cartesian coordinates to polar coordinates.
Let's start by calculating the change in x and y coordinates:
Δx = x_final - x_initial = -4.0 - 3.0 = -7.0
Δy = y_final - y_initial = -2.0 - 2.0 = -4.0
Now, we can calculate the magnitude of the displacement (r) using the Pythagorean theorem:
r = √(Δx² + Δy²) = √((-7.0)² + (-4.0)²) = √(49 + 16) = √65 ≈ 8.06
Next, we need to find the angle (θ) with respect to the positive x-axis. We can use the inverse tangent function:
θ = tan^(-1)(Δy / Δx) = tan^(-1)(-4.0 / -7.0) ≈ 29.74°
Therefore, the displacement in polar coordinates is approximately (8.06, 29.74°).