Question

The cartesian coordinates of a point in the xy plane are x = −5.25 m, y = −2.21 m. Find the distance r from the point to the origin. Answer in units of m. Calculate the angle θ between the radius-vector of the point and the positive x axis (measured counterclockwise from the positive x axis, within the limits of −180◦ to +180◦)

Answer 1

The distance from the point to the origin is approximately 5.69 m. The angle between the radius-vector of the point and the positive x-axis is approximately -22.61°.

Step-by-step explanation:

The distance r from a point to the origin can be found using the Pythagorean theorem. In this case, we have x = -5.25 m and y = -2.21 m. So, the distance r can be calculated as:

r = sqrt(x^2 + y^2) = sqrt((-5.25)^2 + (-2.21)^2) = sqrt(27.56 + 4.8841) = sqrt(32.4441) ≈ 5.69 m

To calculate the angle θ between the radius-vector of the point and the positive x axis, we can use trigonometry. Since the given x and y values are negative, the point is in the third quadrant. Therefore, we need to find the angle in the range of -180° to 0°. We can use the arctangent function to calculate this angle:

θ = atan(y/x) = atan(-2.21/(-5.25)) = atan(0.421) ≈ -22.61°

Answer 2

r = 5.696 and θ = 157.17°.

Step-by-step explanation:

the point is (-5.25,-2.21). The distance from the point to the origin is calculated with the formula

`r = \sqrt{x^(2)+y^(2)} = \sqrt{(-5.25)^(2)+(-2.21)^(2)} = √(27.56+4.88) = √(32.44)`

r = 5.696 m.

I drew the point and the distances in the plane. to find te angle we can use the sin formula:

sin(θ) =
`(2.21)/(r) = (2.21)/(5.696) = 0.388`

θ =
`sin^(-1)(0.388)= 22.83`.

within the range of -180° and 180° the angle will be θ = 180-22.83 = -157.17°