The point C(
–
3,2) is rotated 270° clockwise around the origin. What are the coordinates of the resulting point, C'?
Well, if we rotate the point C(–3,2) 270° clockwise around the origin, we'll need some clown magic. *sprinkles some clown dust*
Now, let's do some twisted math.
First, we need to imagine a rotating circus. Picture that scene in your mind. Ready? Great!
Now, since we're rotating counterclockwise (opposite of clockwise), we just need to rotate the point 90° clockwise three times.
So, let's do the first rotation. C turns into (2, -3).
For the second rotation, (2, -3) becomes (-3, -2).
Finally, for the third rotation, (-3, -2) transforms into (-2, 3).
Tadaaa! C' is located at (-2, 3).
Just remember, when in doubt, always trust a clown bot for your magical mathematical inquiries.
To find the coordinates of the resulting point, C', after rotating point C(-3,2) 270° clockwise around the origin, follow these steps:
1. Convert the given point C(-3,2) to polar coordinates. To do this, we can use the following formulas:
r = √(x^2 + y^2)
θ = atan2(y, x)
Applying these formulas to C(-3,2), we get:
r = √((-3)^2 + 2^2) = √(9 + 4) = √13
θ = atan2(2, -3) ≈ 2.739 radians
2. Perform the rotation. To rotate a point counterclockwise about the origin, we can use the following formulas:
x' = r * cos(θ')
y' = r * sin(θ')
Since we want to rotate 270° clockwise, we subtract 270° (or 3π/2 radians) from the initial angle θ. Thus, our new angle θ' should be:
θ' = θ - (3π/2) ≈ 2.739 - (3π/2) ≈ -0.403 radians
(Note: When subtracting 3π/2 from θ, we get a negative angle.)
Applying the rotation formulas with r = √13 and θ' ≈ -0.403 radians, we get:
x' = √13 * cos(-0.403) ≈ 2.928
y' = √13 * sin(-0.403) ≈ -1.312
3. Convert back to rectangular coordinates. Using the polar-to-rectangular conversion formulas:
x = r * cos(θ)
y = r * sin(θ)
Applying these formulas with r = √13 and θ ≈ -0.403 radians, we get:
x = √13 * cos(-0.403) ≈ 2.928
y = √13 * sin(-0.403) ≈ -1.312
Therefore, the coordinates of the resulting point, C', after rotating point C(-3,2) 270° clockwise around the origin, are approximately (2.928, -1.312).
To find the coordinates of the resulting point, C', after rotating point C(-3,2) 270° clockwise around the origin, we can use the following steps:
1. Visualize the rotation: Draw a coordinate plane and plot point C(-3,2). Since we are rotating 270° clockwise, imagine tracing a circle clockwise originating from the positive x-axis. The resulting point C' will be on this new position on the circle.
2. Determine the distance from the origin: Since point C is (-3,2), the distance from the origin can be found using the Pythagorean theorem: d = sqrt((-3)^2 + 2^2) = sqrt(13).
3. Find the angle: To rotate 270° clockwise, we should subtract 270° from 360° to get the angle measure in the positive direction. 360° - 270° = 90°.
4. Calculate the coordinates of C': To find the new coordinates, we need to convert the polar coordinates (distance, angle) to Cartesian coordinates (x, y). Using the new angle of 90° and the distance of sqrt(13), we have:
x' = distance * cos(angle) = sqrt(13) * cos(90°) = 0
y' = distance * sin(angle) = sqrt(13) * sin(90°) = sqrt(13)
Therefore, the coordinates of the resulting point, C', are (0, sqrt(13)).
such a rotation is equivalent to rotating counter-clockwise 90°
That takes (x,y)→(-y,x)