Rotate the vector (0,2) 270° counter-clockwise about the origin. <[ ? ], [ ]>
The Correct Answer and Explanation is:
<[ 2 ], [ 0 ]>
To rotate a vector in a two-dimensional Cartesian plane, one can use the standard rotation formulas. These formulas describe how the coordinates of a point (x, y) change when rotated counter-clockwise by an angle θ about the origin to a new point (x’, y’). The transformation is given by:
x’ = x cos(θ) – y sin(θ)
y’ = x sin(θ) + y cos(θ)
In this problem, the initial vector is (0, 2), so its coordinates are x = 0 and y = 2. The angle of rotation is θ = 270° counter-clockwise.
The first step is to determine the values of the trigonometric functions for the angle of 270°. On the unit circle, an angle of 270° corresponds to the point (0, -1). Therefore, the cosine of 270° is the x-coordinate, and the sine of 270° is the y-coordinate:
cos(270°) = 0
sin(270°) = -1
Now, these values can be substituted into the rotation formulas along with the coordinates of the original vector.
For the new x-coordinate, x’:
x’ = (0) * cos(270°) – (2) * sin(270°)
x’ = (0)(0) – (2)(-1)
x’ = 0 – (-2)
x’ = 2
For the new y-coordinate, y’:
y’ = (0) * sin(270°) + (2) * cos(270°)
y’ = (0)(-1) + (2)(0)
y’ = 0 + 0
y’ = 0
Thus, the coordinates of the rotated vector are (2, 0).
Geometrically, the original vector (0, 2) points directly upwards along the positive y-axis. A 90° counter-clockwise rotation would place it on the negative x-axis at (-2, 0). A 180° rotation would place it on the negative y-axis at (0, -2). Finally, a 270° counter-clockwise rotation moves it to the positive x-axis. Since the rotation preserves the vector’s magnitude (length), which is 2, the final position must be (2, 0). This confirms the result obtained from the formulas.
