What is the image of the point \( (5,-3) \) after a rotation of \( 90^{\circ} \) counterclockwise about the origin?

To find the image of the point \( (5, -3) \) after a \( 90^{\circ} \) counterclockwise rotation about the origin, we can use the rotation rule. The formula for rotating a point \( (x, y) \) counterclockwise by \( 90^{\circ} \) is \( (-y, x) \). So, for the point \( (5, -3) \): - The new x-coordinate will be \(-(-3) = 3\). - The new y-coordinate will be \(5\). Therefore, the image of the point \( (5, -3) \) after the rotation is \( (3, 5) \). Now, let's add some fun context! Did you know that rotating points in a coordinate plane is a concept that has been explored since ancient times? Ancient Greek mathematicians laid the groundwork for geometry and rotation, studying shapes and their transformations – not just for theory, but also for practical applications in architecture! If you want to dive deeper into how these rotations work, exploring transformations in linear algebra would be an excellent start! Books on vector spaces and matrices often touch upon transformations like rotations, scaling, and translations, illuminating how we can manipulate points in various dimensions. Plus, there are plenty of interactive apps online that let you visualize these transformations in real-time!