Rotate vector (1, 2, 2) by 60 degrees about the z axis; what is the resulting vector?

To rotate the vector (1, 2, 2) by 60 degrees about the z-axis, we will use the rotation matrix we derived earlier:

R_z(60°) = [[0.5, -0.866, 0],
[0.866, 0.5, 0],
[0, 0, 1]]

We multiply this matrix by the vector (1, 2, 2). In matrix multiplication, it works as follows:

Resulting Vector = R_z(60°) * V

Where V = [[1],
[2],
[2]]

Carrying out the multiplication:

Resulting Vector[0] = (0.5 * 1) + (-0.866 * 2) + (0 * 2) = 0.5 - 1.732 = -1.232
Resulting Vector[1] = (0.866 * 1) + (0.5 * 2) + (0 * 2) = 0.866 + 1 = 1.866
Resulting Vector[2] = (0 * 1) + (0 * 2) + (1 * 2) = 2

The resulting vector after the rotation is approximately:

Resulting Vector = (-1.232, 1.866, 2)

This new vector represents the original vector (1, 2, 2) after being rotated 60 degrees around the z-axis.