Previously you found that a 15-degree counterclockwise rotation centered at(2,1)sends the point(4,6)to another point (x, y) ≈ (2.638, 6.347). The diagram on the right shows the vector v in the same direction as vector [2,5], only with a different length. Similarly, the vector u is per- pendicular to v and a different length. Show how you can use (2,1)+v+u to find the rotated point (x,y).
Well, well, well, looks like we're gonna have some fun with vectors! Let's get started, shall we?
We know that the rotation centered at (2, 1) sent the point (4, 6) to (x, y) ≈ (2.638, 6.347). Now, let's bring in our trusty vectors, v and u.
The vector v is in the same direction as [2, 5], just with a different length. Let's say its length is 'm'. Now, since v is in the same direction as [2, 5], we can find its components by multiplying the direction by the magnitude:
v = m * [2, 5]
But wait, there's more! We also have vector u, which is perpendicular to v and of a different length. Let's call its length 'n' (it feels left out).
Now, we can express the rotated point (x, y) as the sum of the initial point (2, 1), vector v, and vector u. So, our equation looks like this:
(x, y) = (2, 1) + v + u
But, we can expand this equation a little further. Since v = m * [2, 5] and u = n * [-5, 2] (remember, u is perpendicular to v), we can replace v and u:
(x, y) = (2, 1) + m * [2, 5] + n * [-5, 2]
Now, all that's left to do is substitute the values of m and n to find the vector that takes our initial point to the rotated point.
I hope that clarifies things a bit! But remember, when it comes to rotations, always keep an eye out for those sneaky clowns! They might just throw a pie in your vector's face! Ha-ha!
To find the rotated point (x, y) using the given vectors (2,5), v, and u, we can follow these steps:
Step 1: Start with the center point (2, 1)
Step 2: Add the vector v, which is in the same direction as (2,5) but a different length. Let's say the length of v is 'a'. So, the new point after adding vector v will be (2,1) + v.
Step 3: Add the vector u, which is perpendicular to v and a different length. Let's say the length of u is 'b'. So, the new point after adding vector u will be (2,1) + v + u.
Step 4: The final point (x, y) will be the result of the previous step, where x is the x-coordinate of the new point and y is the y-coordinate of the new point.
So, the rotated point (x, y) can be found by using the equation (2,1) + v + u.
To find the rotated point (x, y) using (2,1)+v+u, we need to understand the properties of vector addition and how it relates to rotations.
1. Start with the original point (4, 6).
2. Apply a 15-degree counterclockwise rotation centered at (2, 1).
- This rotation involves three steps:
a. Translate the origin to the center of rotation, (2, 1), by subtracting (2, 1) from the original point: (4, 6) - (2, 1) = (2, 5).
b. Perform the rotation on the translated point (2, 5).
c. Translate the origin back to the original position by adding (2, 1) to the result.
3. The vector v is in the same direction as vector [2, 5], with a different length.
4. Similarly, the vector u is perpendicular to v and has a different length.
Now, let's use these steps to find the rotated point (x, y):
1. Start with the original point (4, 6).
2. Translate the origin to the center of rotation, (2, 1).
- Subtract (2, 1) from the original point: (4, 6) - (2, 1) = (2, 5).
3. Apply the 15-degree counterclockwise rotation to the translated point (2, 5).
- To rotate a point counterclockwise, we can use the following rotation matrix:
R = | cos(θ) -sin(θ) |
| sin(θ) cos(θ) |
In this case, θ = 15 degrees.
So, we can express the rotation as:
x' = cos(15°) * x - sin(15°) * y
y' = sin(15°) * x + cos(15°) * y
Substituting the values of (x, y) = (2, 5) into the above equations:
x' = cos(15°) * 2 - sin(15°) * 5
y' = sin(15°) * 2 + cos(15°) * 5
4. Translate the origin back to the original position.
- Add (2, 1) to the resulting point (x', y') obtained from the rotation.
- (x, y) = (x', y') + (2, 1)
x = (cos(15°) * 2 - sin(15°) * 5) + 2 ≈ 2.638
y = (sin(15°) * 2 + cos(15°) * 5) + 1 ≈ 6.347
Therefore, the rotated point (x, y) is approximately (2.638, 6.347) when using the vector addition (2,1)+v+u.
The unit vector a in the direction of v is a=v/|v|
similarly for u, b=u/|u|
Now, let w be the vector from (2,1) to the new point. Then
w = (2.638, 6.347)-(2,1) = [0.638,5.347]
Draw a right triangle, such that
k(a+b) = w
so, (2,1)+k(v/|v| + u/|u|) = (2.638, 6.347)