Two points in the xy plane have Cartesian coordinates (2.00, -4.00) m and (-5.00, 5.00) m.
(a)What is the distance between these two points?
(b) What are their polar coordinates?
To find the distance between two points in the Cartesian coordinate system, you can use the distance formula:
(a) Distance between two points = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's substitute the given values:
x1 = 2.00 m, y1 = -4.00 m
x2 = -5.00 m, y2 = 5.00 m
Using the formula:
Distance = sqrt((-5.00 - 2.00)^2 + (5.00 - (-4.00))^2)
Distance = sqrt((-7.00)^2 + (9.00)^2)
Distance = sqrt(49.00 + 81.00)
Distance = sqrt(130.00)
Distance ≈ 11.40 m
(b) To find the polar coordinates of a point, you can use the following formulas:
r = sqrt(x^2 + y^2)
θ = tan^(-1)(y/x)
Let's substitute the given values:
x = -5.00 m, y = 5.00 m
Using the formulas:
r = sqrt((-5.00)^2 + (5.00)^2)
r = sqrt(25.00 + 25.00)
r = sqrt(50.00)
r ≈ 7.07 m
θ = tan^(-1)(5.00/-5.00)
θ = tan^(-1)(-1.00)
θ ≈ -45.00°
Therefore, the polar coordinates of the second point (-5.00, 5.00) in the xy plane are approximately (7.07 m, -45.00°).
To find the distance between two points in the Cartesian coordinate plane, you can use the distance formula:
(a) Distance between two points:
The distance formula is derived from the Pythagorean theorem. For two points with coordinates (x1, y1) and (x2, y2), the distance is given by:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, the coordinates of the two points are:
Point 1 -> (2.00, -4.00)
Point 2 -> (-5.00, 5.00)
Using the formula, substituting the coordinates into the formula, we get:
Distance = sqrt((-5.00 - 2.00)^2 + (5.00 - (-4.00))^2)
Calculating the values within the parentheses:
Distance = sqrt((-7.00)^2 + (9.00)^2)
Simplifying:
Distance = sqrt(49.00 + 81.00)
Distance = sqrt(130.00)
Distance ≈ 11.40 meters
Therefore, the distance between the two points is approximately 11.40 meters.
(b) Polar coordinates:
To find the polar coordinates of a point, we need to express the point in terms of its distance from the origin (radius) and its angle from the positive x-axis (theta).
The formula to convert Cartesian coordinates (x, y) to polar coordinates (r, θ) is:
r = sqrt(x^2 + y^2)
θ = arctan(y/x)
Using the coordinates of Point 1 -> (2.00, -4.00):
r = sqrt(2.00^2 + (-4.00)^2)
r = sqrt(4.00 + 16.00)
r = sqrt(20.00)
r ≈ 4.47 meters
θ = arctan((-4.00)/2.00)
θ = arctan(-2.00)
θ ≈ -63.43 degrees
Therefore, the polar coordinates of Point 1 are approximately (4.47 m, -63.43 degrees).
Using the coordinates of Point 2 -> (-5.00, 5.00):
r = sqrt((-5.00)^2 + (5.00)^2)
r = sqrt(25.00 + 25.00)
r = sqrt(50.00)
r ≈ 7.07 meters
θ = arctan((5.00)/(-5.00))
θ = arctan(-1.00)
θ ≈ -45.00 degrees
Therefore, the polar coordinates of Point 2 are approximately (7.07 m, -45.00 degrees).
Difference in x = -5.00 -2.00 = -7
Difference in y = 5.00 - -4.00 = 9
hypotenuse = sqrt (49 +81) = sqrt(130)=11.40
so the distance is 11.40 meters
graph (2,-4)
that is in quadrant 4
it is 4 units below the x axis and two o the right of the y axis
tangent of angle below x axis = -4/2 = -2
tan^-1(2) = 63.4
so 63.4 degrees below x axis or 360-63.4 counterclockwise from x axis
so theta = 296.6 degrees or 5.18 radians
r = sqrt (2^2 + -4^2) = sqrt 20 = 2sqrt5
so first point in r theta is
r = 2 sqrt 5
theta = 5.18 radians or 296.6 degrees
Find r, theta for the second point the same way.
Theta =tan^5/-5=-45
R=Seq5+5=7.0