To rotate the point \( P (1, -4) \) by \( 270^\circ \) clockwise, we can first convert this rotation into a more manageable form. A \( 270^\circ \) clockwise rotation is equivalent to a \( 90^\circ \) counterclockwise rotation.
The general formula for rotating a point \( (x, y) \) by \( 90^\circ \) counterclockwise is:
\[ (x, y) \rightarrow (-y, x) \]
Let's apply this formula step-by-step:
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Identify the coordinates of point \( P \):
- \( x = 1 \)
- \( y = -4 \)
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Apply the rotation formula: \[ P' = (-y, x) = (-(-4), 1) = (4, 1) \]
So, the coordinates of point \( P' \) after a \( 270^\circ \) clockwise rotation of point \( P (1, -4) \) are:
\[ \boxed{(4, 1)} \]
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