Apply the 57-degree counterclockwise rotation about the origin to the vectors
[1, 0] and [0,1], then use the image vectors (written as columns) to form the coe cient matrix M for the rotation. Test M by calculating the products
M[1]
[0] and
M[0]
[1 ]. Where does this
rotation send the vector [3, 1]? Does M, when applied to
[3]
[1] , do its job correctly?
btw...the "M's" mean matrix
To apply a 57-degree counterclockwise rotation about the origin to the vectors [1, 0] and [0, 1], we can use trigonometric functions.
To rotate the vector [1, 0], we can use the following formula:
x' = cos(theta) * x - sin(theta) * y
y' = sin(theta) * x + cos(theta) * y
Substituting the values, we get:
x' = cos(57) * 1 - sin(57) * 0 = cos(57)
y' = sin(57) * 1 + cos(57) * 0 = sin(57)
Similarly, for the vector [0, 1], we get:
x' = cos(57) * 0 - sin(57) * 1 = -sin(57)
y' = sin(57) * 0 + cos(57) * 1 = cos(57)
So, the image vectors (written as columns) are:
[cos(57)] [-sin(57)]
[sin(57)] [cos(57)]
Using these image vectors as columns, we can form the coefficient matrix, which we'll call M:
M = [cos(57), -sin(57)]
[sin(57), cos(57)]
To test M by calculating the products M[1] and M[0], we multiply M by the vectors [1] and [0]. We get:
M[1] = [cos(57), -sin(57)] * [1] = [cos(57)]
[sin(57), cos(57)] [0] [sin(57)]
M[0] = [cos(57), -sin(57)] * [0] = [-sin(57)]
[sin(57), cos(57)] [1] [cos(57)]
Now, let's find where this rotation sends the vector [3, 1] by multiplying it with M:
[cos(57), -sin(57)] * [3] = [3*cos(57) - sin(57)]
[sin(57), cos(57)] [1] [sin(57) + cos(57)]
So, the rotated vector is [3*cos(57) - sin(57), sin(57) + cos(57)].
To check if M, when applied to [3] and [1], does its job correctly, we can compare the computed rotated vector [3*cos(57) - sin(57), sin(57) + cos(57)] with the expected result.
Note: The numerical values for cos(57) and sin(57) can be calculated using a calculator or software.
To apply a counterclockwise rotation about the origin to a vector, we can use the following rotation matrix:
M = [[cosθ, -sinθ],
[sinθ, cosθ]]
where θ is the angle of rotation in radians. In this case, the angle of rotation is 57 degrees, which needs to be converted to radians.
θ = 57 degrees * (π/180) ≈ 0.9948 radians
Now, let's apply the rotation to the vectors [1, 0] and [0, 1]:
Applying M to [1, 0]:
M[1] = [cosθ, -sinθ] * [1] = [cosθ] = [0.5403]
M[0] [sinθ, cosθ] [0] [sinθ] [0.8415]
Applying M to [0, 1]:
M[0] = [cosθ, -sinθ] * [0] = [-sinθ] = [-0.8415]
M[1] [sinθ, cosθ] [1] [cosθ] [0.5403]
Now, let's form the coefficient matrix M for the rotation:
M = [[0.5403, -0.8415],
[0.8415, 0.5403]]
To test the matrix, let's calculate the product of M and the vector [3, 1]:
M[3] = [0.5403, -0.8415] * [3] = [-0.4450] ≈ [-0.445]
M[1] [0.8415, 0.5403] [1] [ 2.3245] [ 2.325]
So, when we apply the rotation matrix M to the vector [3, 1], it results in the vector approximately equal to [-0.445, 2.325].
M does its job correctly as it rotates the vector [3, 1] counterclockwise around the origin by 57 degrees.