Question
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What is the image point of $(0,7)$ after the transformation $R_{270} \circ r_{y=-x}$ ?
Answer
Final Answer: The image point of \((0,7)\) after the transformation \(R_{270} \circ r_{y=-x}\) is \(\boxed{(0,7)}\).
Step 1 ：The transformation \(R_{270} \circ r_{y=-x}\) means that we first reflect the point across the line \(y=-x\) and then rotate the result 270 degrees counterclockwise about the origin.
Step 2 ：To reflect a point across the line \(y=-x\), we simply swap the x and y coordinates and change their signs. So the image of \((0,7)\) under the reflection \(r_{y=-x}\) is \((-7,0)\).
Step 3 ：Next, to rotate a point 270 degrees counterclockwise about the origin, we can use the rotation matrix for 270 degrees, which is \(\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\). Multiplying this matrix by the column vector of our point will give us the coordinates of the image point.
Step 4 ：Performing the matrix multiplication, we find that the new point is \((0,7)\).
Step 5 ：Final Answer: The image point of \((0,7)\) after the transformation \(R_{270} \circ r_{y=-x}\) is \(\boxed{(0,7)}\).
