Question

the point v(4, -1) is rotated 90° clockwise around the origin. what are the coordinates of its image v? 〇 v(1, 4) 〇 v(-1, 4) 〇 v(-1, -4) 〇 v(4, 1)

Explanation:

Step1: Recall 90° clockwise rotation rule

For a point $(x,y)$, 90° clockwise rotation around the origin transforms it to $(y, -x)$.

Step2: Substitute $x=4, y=-1$

Substitute into the rule: $(y, -x) = (-1, -4)$? No, correct substitution: $(y, -x) = (-1, -4)$ is wrong, correct: $(y, -x) = (-1, - (4))$? No, correct rule: 90° clockwise rotation is $(x,y) \to (y, -x)$. So $x=4$, $y=-1$: new $x$ is $y=-1$, new $y$ is $-x=-4$? No, wait, correct rule: 90° clockwise around origin: $(x,y) \to (y, -x)$. So $(4,-1) \to (-1, -4)$? No, wait no, let's correct: 90° clockwise rotation formula is $(x,y) \mapsto (y, -x)$. So $x=4$, $y=-1$: new coordinate is $(-1, -4)$? No, wait no, another way: 90° clockwise is equivalent to 270° counterclockwise, formula $(x,y) \to (y, -x)$. So $(4,-1)$: $y=-1$, $-x=-4$, so $(-1, -4)$? Wait no, let's plot: original point (4,-1) is in 4th quadrant. Rotate 90° clockwise around origin, it moves to 3rd quadrant? No, wait 4th quadrant, rotate 90° clockwise: down to 3rd? No, 4th quadrant: x positive, y negative. Rotate 90° clockwise: the x becomes original y, y becomes -original x. So $(4,-1) \to (-1, -4)$. Wait but let's check options: V'(-1,-4) is an option. Wait no, wait maybe I mixed up the rule. Wait 90° counterclockwise is $(x,y) \to (-y,x)$, 90° clockwise is $(x,y) \to (y,-x)$. Yes, so (4,-1): y is -1, -x is -4, so (-1, -4). Wait but let's verify with another method: the vector from origin to (4,-1) is $\vec{v} =

$$\begin{pmatrix}4\\-1\end{pmatrix}$$

$. The rotation matrix for 90° clockwise is

$$\begin{pmatrix}0&1\\-1&0\end{pmatrix}$$

? No, wait rotation matrix for $\theta$ clockwise is

$$\begin{pmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{pmatrix}$$

. $\theta=90°$, $\cos90=0$, $\sin90=1$. So matrix is

$$\begin{pmatrix}0&1\\-1&0\end{pmatrix}$$

. Multiply by

$$\begin{pmatrix}4\\-1\end{pmatrix}$$

: $0*4 +1*(-1) = -1$, $-1*4 +0*(-1) = -4$. So result is

$$\begin{pmatrix}-1\\-4\end{pmatrix}$$

.

Answer:

V'(-1, -4)