To rotate the given triangle \(90^\circ\) clockwise about the origin, we will follow these steps:
1. Understand the initial points: The triangle is represented by the coordinates in the matrix:
[tex]\[
\left[\begin{array}{ccc}
0 & -3 & 5 \\
0 & 1 & 2
\end{array}\right]
\][/tex]
Each column represents a vertex of the triangle. So, we have the vertices \((0, 0)\), \((-3, 1)\), and \((5, 2)\).
2. Understand the rotation: A \(90^\circ\) clockwise rotation about the origin will transform any point \((x, y)\) to \((y, -x)\).
3. Apply the transformation to each point:
- Vertex \((0, 0)\) becomes \((0, 0)\).
- Vertex \((-3, 1)\) becomes \((1, 3)\).
- Vertex \((5, 2)\) becomes \((2, -5)\).
4. Compile the rotated vertices into a matrix:
[tex]\[
\left[\begin{array}{ccc}
0 & 1 & 2 \\
0 & 3 & -5
\end{array}\right]
\][/tex]
So, the coordinates of the rotated triangle vertices are \((0, 0)\), \((1, 3)\), and \((2, -5)\). This gives us the final matrix representing the rotated triangle:
[tex]\[
\left[\begin{array}{ccc}
0 & 1 & 2 \\
0 & 3 & -5
\end{array}\right]
\][/tex]
