Let F (x) = `[{:(cos x , -sin x , 0),(sin x , cos x , 0),(0 , 0, 1):}]`, then

Options

A. F (x). F(y) = F (x + y)
B. F(x)F(y) = F (x). F (y)
C. F(x) . F (y) = F(x - y )
D. F(x) . F(y) = F(y - x)

Answer

To solve the problem, we need to find the product of the matrices \( F(x) \) and \( F(y) \) and simplify the result. Let's go through the steps systematically.

### Step 1: Define the matrices
Given:
\[ F(x) = \begin{pmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{pmatrix} \]
To find \( F(y) \), we replace \( x \) with \( y \):
\[ F(y) = \begin{pmatrix} \cos y & -\sin y & 0 \\ \sin y & \cos y & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

### Step 2: Multiply \( F(x) \) and \( F(y) \)
We need to compute the product \( F(x) \cdot F(y) \):
\[ F(x) \cdot F(y) = \begin{pmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} \cos y & -\sin y & 0 \\ \sin y & \cos y & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

### Step 3: Calculate the elements of the resulting matrix
We will calculate each element of the resulting matrix using the row-by-column multiplication method.

1. **First Row:**
- First column:
\[ \cos x \cdot \cos y + (-\sin x) \cdot \sin y + 0 \cdot 0 = \cos x \cos y - \sin x \sin y \]
- Second column:
\[ \cos x \cdot (-\sin y) + (-\sin x) \cdot \cos y + 0 \cdot 0 = -\cos x \sin y - \sin x \cos y \]
- Third column:
\[ \cos x \cdot 0 + (-\sin x) \cdot 0 + 0 \cdot 1 = 0 \]

2. **Second Row:**
- First column:
\[ \sin x \cdot \cos y + \cos x \cdot \sin y + 0 \cdot 0 = \sin x \cos y + \cos x \sin y \]
- Second column:
\[ \sin x \cdot (-\sin y) + \cos x \cdot \cos y + 0 \cdot 0 = -\sin x \sin y + \cos x \cos y \]
- Third column:
\[ \sin x \cdot 0 + \cos x \cdot 0 + 0 \cdot 1 = 0 \]

3. **Third Row:**
- First column:
\[ 0 \cdot \cos y + 0 \cdot \sin y + 1 \cdot 0 = 0 \]
- Second column:
\[ 0 \cdot (-\sin y) + 0 \cdot \cos y + 1 \cdot 0 = 0 \]
- Third column:
\[ 0 \cdot 0 + 0 \cdot 0 + 1 \cdot 1 = 1 \]

### Step 4: Combine the results
Putting all the calculated elements together, we get:
\[ F(x) \cdot F(y) = \begin{pmatrix} \cos x \cos y - \sin x \sin y & -(\cos x \sin y + \sin x \cos y) & 0 \\ \sin x \cos y + \cos x \sin y & \cos x \cos y - \sin x \sin y & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

### Step 5: Simplify using trigonometric identities
Using the trigonometric identities:
- \( \cos(x+y) = \cos x \cos y - \sin x \sin y \)
- \( \sin(x+y) = \sin x \cos y + \cos x \sin y \)

We can rewrite the matrix as:
\[ F(x) \cdot F(y) = \begin{pmatrix} \cos(x+y) & -\sin(x+y) & 0 \\ \sin(x+y) & \cos(x+y) & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

### Conclusion
Thus, we conclude that:
\[ F(x) \cdot F(y) = F(x+y) \]