Question

If we have a point at (5,3) and we rotate this point about the origin by -30 degrees with respect to the x axis. what is it's new location?

A) (-3.5, 4.598)
B) (4.598, -3.5)
C) (-4.598, 3.5)
D) (3.5, -4.598)

Answer 1

After rotating the point (5,3) about the origin by -30 degrees, the new location of the point is approximately (4.598, 2.098) using the rotation matrix. The option that matches closest to these calculations is not listed, indicating a possible error in the given solutions.

Step-by-step explanation:

The student is asking about the new location of a point after it has been rotated about the origin. To calculate the new coordinates of the point (5,3) after a rotation of -30 degrees, we can use the rotation matrix.

To apply a rotation by an angle θ in a counterclockwise direction, the rotation matrix is as follows:

[ cos(θ) -sin(θ) sin(θ) cos(θ)]

For a rotation of -30 degrees (which is equivalent to 30 degrees in the clockwise direction), we use a positive angle in the matrix because the clockwise direction is negative as per convention.

So, the rotation matrix for -30 degrees is:

[ cos(30°) sin(30°)
-sin(30°) cos(30°)]

Using these values:

cos(30°) = √3/2
sin(30°) = 1/2So our rotation matrix is:

[ √3/2 1/2
-1/2 √3/2]

Applying this matrix to the point (5,3) we get:

x' = 5 * (√3/2) + 3 * (1/2) = 4.598
y' = 5 * (-1/2) + 3 * (√3/2) = -0.5 + 2.598 = 2.098

Therefore, the new location of the point after rotation is approximately (4.598, 2.098). This is close to one of the answers provided, but since the exact match is not there, either the question made a rounding error or the answer choices do not have the correct option.