Point P(x, y) is rotated by an angle `theta` in anticlockwise direction. The new position of point P is `Q (x_(1), y_(1))`. If `[(x_(1)),(y_(1))]=A[(x),(y)]`, then find matrix A.

Explanation

To find the matrix \( A \) that represents the rotation of point \( P(x, y) \) to point \( Q(x_1, y_1) \) by an angle \( \theta \) in an anticlockwise direction, we can follow these steps:

### Step 1: Understand the Rotation
When a point \( P(x, y) \) is rotated by an angle \( \theta \), its new coordinates \( Q(x_1, y_1) \) can be expressed in terms of the original coordinates \( (x, y) \). The formulas for the new coordinates after rotation are:
\[ x_1 = x \cos \theta - y \sin \theta \]
\[ y_1 = x \sin \theta + y \cos \theta \]

### Step 2: Write the Equations in Matrix Form
We can express the above equations in matrix form. This can be done by writing:
\[ \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \]

### Step 3: Identify Matrix \( A \)
From the matrix equation above, we can identify matrix \( A \) as:
\[ A = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \]

### Conclusion
Thus, the required matrix \( A \) that represents the rotation of point \( P(x, y) \) to point \( Q(x_1, y_1) \) is:
\[ A = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \]

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