The cartesian coordinates of a point in the xy plane are x= -3.64 m, y = -1.75 m.
1. Find the distance, r, from the point to the origin. Answer in units of m.
2. Calculate the angle between the radius-vector of the point and the positive x axis (measured counterclockwise from the positive x axis, within the limits of -180 degrees to +180 degrees). Answer in units of degrees.
1. Use the Pythagorean Theorem
sqrt[(3.64)^2 + (1.75)^2] = ___
2. The angle is in the third quadrant. The angle or the radius vector with the -x axis is
arctan 1.75/3.64 = 25.7 degrees
The angle is 205.7 degrees, but they want it expressed as between -180 and +180, so that would be -154.3 degrees
To find the distance, r, from the point to the origin, you can use the Pythagorean theorem. The Pythagorean theorem states that for a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the x-coordinate of the point is -3.64 m and the y-coordinate is -1.75 m. Therefore, the distance, r, can be calculated as follows:
r = sqrt((-3.64)^2 + (-1.75)^2)
r = sqrt(13.2796 + 3.0625)
r = sqrt(16.3421)
r ≈ 4.04 m
So, the distance from the point to the origin is approximately 4.04 m.
To calculate the angle between the radius-vector of the point and the positive x-axis, you can use the inverse tangent function (also known as arctan or atan). The inverse tangent function takes the ratio of the y-coordinate to the x-coordinate and gives you the angle in radians.
In this case, the y-coordinate is -1.75 m and the x-coordinate is -3.64 m. Therefore, the angle can be calculated as follows:
angle = atan(-1.75 / -3.64)
angle ≈ atan(0.4808)
angle ≈ 25.88 degrees
However, this angle is measured counterclockwise from the positive x-axis. To get the angle within the limits of -180 degrees to +180 degrees, you can subtract it from 180 degrees for angles in the second and third quadrants, or add it to 360 degrees for angles in the fourth quadrant.
In this case, the angle is in the second quadrant, so you subtract it from 180 degrees:
angle ≈ 180 - 25.88
angle ≈ 154.12 degrees
So, the angle between the radius-vector of the point and the positive x-axis is approximately 154.12 degrees.