Previously you found that a 15-degree counterclockwise rotationcenteredat(2,1)sendsthepoint(4,6)toanother 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 di↵erent length. Similarly, the vector u is per- pendicular to v and a di↵erent length. Show how you can use (2,1)+v+u to find the rotated point (x,y).
Yesit'scertainlyweird,butthequestionisquiteinteresting!
Let
C=(2,1) [centre of rotation]
P=(4,6) [point to be rotated]
t=π/12 [15°, angle of rotation]
The first part of the problem is solved as follows:
Rotated point, P', is obtained by
1. translating the point P so that C becomes the origin,
P->(P-C)
2. roatate by π/12 clockwise
multiply by the rotation matrix, R(t):
| cos(t) -sin(t) |
| sin(t) cos(t) |
P->R(t)P
3. reverse the translation
P->P+C
Summing up, the rotation procedure is
P'=C+R(t)(P-C)
=(2cos(π/12)-5sin(π/12)+2, 5cos(%pi;/12)+2sin(π/12)+1)
Second part of the question.
It turns out that the vector <2,5> is exactly the vector
P-C=<4-2,6-1>=<2,5>
and it turns out that the two rows of the rotation matrix
<cos(t), -sin(t)> and
<sin(t), cos(t)>
are orthogonal (perpendicular).
This means that
R(t)<P-C>=
|2 5| <cos(t),-sin(t)>
|5 -2|
=cos(t)<2,5>-sin(t)<5,-2>
=u+v
By translating back, we get
(2,1)+u+v
sorry it's formatted weirdly
To find the rotated point (x, y) using the expression (2,1)+v+u, we need to determine the values of vectors v and u.
1. Vector v:
- Vector v is in the same direction as vector [2, 5] but with a different length.
- Given that the point (4, 6) is rotated counterclockwise by 15 degrees centered at (2, 1) to (2.638, 6.347), we can find the vector v that represents this rotation.
- The translation from the center of rotation to the original point is (4 - 2, 6 - 1) = (2, 5).
- To find vector v, we need to rotate this translation by 15 degrees counterclockwise.
- Multiply the original vector by the rotation matrix:
[ cosθ -sinθ ] [ 2 ] [ cos(15°) -sin(15°) ] [ 2 ] [ 0.9659 -0.2588 ] [ 2 ] [ 1.9317 -0.5176 ]
[ sinθ cosθ ] * [ 5 ] = [ sin(15°) cos(15°) ] * [ 5 ] = [ 0.2588 0.9659 ] * [ 5 ] = [ 1.294 4.8295 ]
Therefore, vector v = [1.9317, -0.5176].
2. Vector u:
- Vector u is perpendicular to vector v and has a different length.
- Since u is perpendicular to v, the dot product of u and v is zero (u · v = 0).
- Let's assume that vector u = [x, y].
- Using the dot product condition, we have:
(1.9317 * x) + (-0.5176 * y) = 0 (Equation 1)
- Vector u can have any length, so we can choose a specific value for either x or y to simplify the equation.
- Let's choose x = 1, then the equation becomes:
(1.9317 * 1) + (-0.5176 * y) = 0
1.9317 - 0.5176 * y = 0
-0.5176 * y = -1.9317
y = -1.9317 / -0.5176
y ≈ 3.728
Therefore, vector u ≈ [1, 3.728].
3. Finding the rotated point (x, y):
- Using the expression (2,1)+v+u, we add the vectors v and u to the point (2,1):
(2, 1) + (1.9317, -0.5176) + (1, 3.728)
= (2 + 1.9317 + 1, 1 - 0.5176 + 3.728)
= (4.9317, 4.2114)
Therefore, the rotated point (x, y) is approximately (4.9317, 4.2114).
To find the rotated point (x, y) using the vector v and u, we can follow these steps:
1. Start with the center of rotation, which is (2, 1).
2. Add the vector v to the center of rotation. The vector v has the same direction as [2, 5] but with a different length. This gives us a new point, let's call it point A.
- If the length of vector v is known, you can simply multiply the components of [2, 5] by that length. For example, if the length of v is 3, then A = (2, 1) + 3[2, 5] = (2, 1) + [6, 15] = (8, 16).
- If the length of vector v is not given, but its scaled position is given, like in this case where point A is given as (2.638, 6.347), you can subtract the center of rotation from point A to get the vector v. For example, v = A - (2, 1) = (2.638, 6.347) - (2, 1) = (0.638, 5.347).
3. Add the vector u to point A. The vector u is perpendicular to v and has a different length. This gives us the final rotated point (x, y).
- If the length of vector u is known, you can simply add its components to point A. For example, if the length of u is 2, then (x, y) = A + 2u.
- If the length of vector u is not given, but its scaled position is given, you can subtract point A from the final rotated point (x, y) to get the vector u. For example, (x, y) - A = u.
By following these steps and knowing the appropriate values for the vectors v and u, you can find the rotated point (x, y) using the equation (2, 1) + v + u.