Matrices & Determinants
Matrix Operations
Grade 12
Question:
<p>Point <em>P</em>(<em>x</em>, <em>y</em>) is rotated by an angle <em>θ</em> in anticlockwise direction. The new position of point <em>P</em> is <em>Q</em>(<em>x</em><sub>1</sub>, <em>y</em><sub>1</sub>). If \(\begin{bmatrix} x_1 \\ y_1 \end{bmatrix} = A \begin{bmatrix} x \\ y \end{bmatrix}\), then find matrix <em>A</em>.</p>
<p>\(A = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}\)</p>
<p>\(A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}\)</p>
<p>\(A = \begin{bmatrix} \sin\theta & \cos\theta \\ \cos\theta & -\sin\theta \end{bmatrix}\)</p>
<p>\(A = \begin{bmatrix} -\cos\theta & \sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}\)</p>
Step-by-Step Solution
Key Concept: The rotation transformation matrix for anticlockwise rotation by angle θ is a standard linear transformation where the new coordinates are expressed as a matrix product of the rotation matrix with the original position vector.
<p><strong>Step 1:</strong> Set up the rotation transformation. When point P(x, y) is rotated anticlockwise by angle θ about the origin, the new coordinates (x₁, y₁) are related to the original coordinates by the rotation matrix.</p><p><strong>Step 2:</strong> Use the standard rotation transformation formulas:</p><p>x₁ = x cos θ - y sin θ</p><p>y₁ = x sin θ + y cos θ</p><p><strong>Step 3:</strong> Express in matrix form:</p><p>$$\begin{bmatrix} x_1 \\ y_1 \end{bmatrix} = \begin{bmatrix} \cos θ & -\sin θ \\ \sin θ & \cos θ \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$$</p><p><strong>Step 4:</strong> Therefore, matrix A is the anticlockwise rotation matrix:</p><p>$$A = \begin{bmatrix} \cos θ & -\sin θ \\ \sin θ & \cos θ \end{bmatrix}$$</p><p>∴ Answer: B</p>
Correct Answer: B