To determine the location of the vertex \(Q\) of the original rectangle given its image \( Q^{\prime} \) at \((-3, 4)\) after a \(90^\circ\) clockwise rotation around the origin, we follow these steps:
1. Understand the Transformation:
- A \(90^\circ\) clockwise rotation around the origin takes a point \((x, y)\) to \((y, -x)\).
2. Reverse the Rotation:
- To find the original coordinates of \( Q \) before the transformation, we need to reverse the \(90^\circ\) clockwise rotation.
- The reverse of a \(90^\circ\) clockwise rotation is a \(90^\circ\) counterclockwise rotation.
3. Apply the Reverse Transformation:
- For a \(90^\circ\) counterclockwise rotation, a point \((x', y')\) transforms to \((-y', x')\).
4. Calculate the Original Coordinates:
- We have \( Q^{\prime} \) at \((-3, 4)\).
- Applying the reverse transformation:
[tex]\[
x = 4 \quad \text{(formerly y')}
\][/tex]
[tex]\[
y = -(-3) = 3 \quad \text{(formerly -x')}
\][/tex]
Therefore, the location of \( Q \) is:
[tex]\[
\boxed{(4, 3)}
\][/tex]
