To rotate a point clockwise, we need to follow a specific process. Let’s break it down step by step:
1. Determine the coordinates of the point you want to rotate. Let’s say we have a point P(x, y).
2. Identify the point about which you want to rotate the given point. This point is often referred to as the center of rotation. Let’s call it C(a, b).
3. Calculate the distance between the point to be rotated and the center of rotation using the distance formula:
D = √((x – a)^2 + (y – b)^2)
4. Find the angle of rotation. In this case, since we want to rotate the point clockwise, the angle of rotation will be negative. Let’s call it θ.
5. Apply the rotation formula to find the new coordinates of the point. The formula is as follows:
X’ = a + (x – a)cos(θ) – (y – b)sin(θ)
Y’ = b + (x – a)sin(θ) + (y – b)cos(θ)
Plugging in the values, we get:
X’ = a + (x – a)cos(-θ) – (y – b)sin(-θ)
Y’ = b + (x – a)sin(-θ) + (y – b)cos(-θ)
Simplifying further, we have:
X’ = a + (x – a)cos(θ) + (y – b)sin(θ)
Y’ = b – (x – a)sin(θ) + (y – b)cos(θ)
6. Substitute the values of x, y, a, b, and θ into the above formulas to calculate the new coordinates of the rotated point.
It’s important to note that the angle of rotation, θ, is measured in radians. If you are given the angle in degrees, you can convert it to radians by multiplying it by π/180.
Let’s illustrate this process with an example:
Suppose we want to rotate the point P(3, 4) 90 degrees clockwise about the center of rotation C(1, 1).
1. The coordinates of the point P are x = 3 and y = 4.
2. The center of rotation is C(1, 1).
3. The distance between P and C can be calculated as:
D = √((3 – 1)^2 + (4 – 1)^2)
= √(2^2 + 3^2)
= √(4 + 9)
= √13
4. The angle of rotation is 90 degrees, which is equivalent to π/2 radians.
5. Applying the rotation formulas, we get:
X’ = 1 + (3 – 1)cos(π/2) + (4 – 1)sin(π/2)
= 1 + 2(0) + 3(1)
= 1 + 0 + 3
= 4
Y’ = 1 – (3 – 1)sin(π/2) + (4 – 1)cos(π/2)
= 1 – 2(1) + 3(0)
= 1 – 2 + 0
= -1
Therefore, the new coordinates of the rotated point P’ are (4, -1).
By following these steps, you can rotate any point clockwise about a given center of rotation.