To find the image of the point \( V(-5, -1) \) after a rotation of 270° counterclockwise around the origin, we can use the rotation transformation formula for rotating a point \((x, y)\) counterclockwise by an angle \(\theta\):
\[ (x', y') = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta) \]
For a rotation of 270°, we have:
- \( \cos(270^\circ) = 0 \)
- \( \sin(270^\circ) = -1 \)
Now, apply the rotation formula to the point \((-5, -1)\):
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Substitute the values: \[ x = -5, \quad y = -1 \]
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Calculate \(x'\) and \(y'\): \[ x' = -5 \cos(270^\circ) - (-1) \sin(270^\circ) = -5(0) - (-1)(-1) = 0 - 1 = -1 \]
\[ y' = -5 \sin(270^\circ) + (-1) \cos(270^\circ) = -5(-1) + (-1)(0) = 5 + 0 = 5 \]
Putting \(x'\) and \(y'\) together, the image of the point \( V(-5, -1) \) after a rotation of 270° counterclockwise around the origin is:
\[ V'(-1, 5) \]
Thus, the final answer is the new coordinates after the rotation:
Image: \( V'(-1, 5) \)