Questions: The cartesian coordinates of a point in the xy plane are x=-5.17 m, y=-3.85 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 in units of °.

Solution Steps

To find the distance \( r \) from the point \((-5.17 \, \text{m}, -3.85 \, \text{m})\) to the origin, we use the distance formula: \[ r = \sqrt{x^2 + y^2} \] Substituting the given values: \[ r = \sqrt{(-5.17)^2 + (-3.85)^2} \] \[ r = \sqrt{26.7289 + 14.8225} \] \[ r = \sqrt{41.5514} \] \[ r \approx 6.4482 \, \text{m} \]

\[ \boxed{r \approx 6.4482 \, \text{m}} \]

To find the angle \( \theta \) between the radius vector of the point and the positive \( x \) axis, we use the arctangent function: \[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \] Substituting the given values: \[ \theta = \tan^{-1}\left(\frac{-3.85}{-5.17}\right) \] \[ \theta = \tan^{-1}(0.7447) \] \[ \theta \approx 36.57^\circ \] Since both \( x \) and \( y \) are negative, the point is in the third quadrant. Therefore, we need to add \( 180^\circ \) to the angle to get the correct direction: \[ \theta = 36.57^\circ - 180^\circ \] \[ \theta \approx -143.43^\circ \]

\[ \boxed{\theta \approx -143.43^\circ} \]

Final Answer