To solve the problem, we need to find the equation of the reflected ray after a ray of light strikes a plane mirror defined by three points. Let's go through the solution step by step.
### Step 1: Determine the Equation of the Plane
The three points given are \( A(2, 1, 1) \), \( B(3, 0, 2) \), and \( C(2, -1, -2) \). We can find the normal vector of the plane using the cross product of two vectors formed by these points.
1. **Find vectors AB and AC:**
\[
\vec{AB} = B - A = (3 - 2, 0 - 1, 2 - 1) = (1, -1, 1)
\]
\[
\vec{AC} = C - A = (2 - 2, -1 - 1, -2 - 1) = (0, -2, -3)
\]
2. **Compute the cross product \( \vec{n} = \vec{AB} \times \vec{AC} \):**
\[
\vec{n} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
1 & -1 & 1 \\
0 & -2 & -3
\end{vmatrix}
= \hat{i}((-1)(-3) - (1)(-2)) - \hat{j}((1)(-3) - (1)(0)) + \hat{k}((1)(-2) - (-1)(0))
\]
\[
= \hat{i}(3 + 2) - \hat{j}(-3) + \hat{k}(-2)
\]
\[
= 5\hat{i} + 3\hat{j} - 2\hat{k}
\]
3. **Equation of the plane:**
The equation of the plane can be written as:
\[
5(x - 2) + 3(y - 1) - 2(z - 1) = 0
\]
Simplifying this gives:
\[
5x + 3y - 2z - 4 = 0
\]
### Step 2: Identify the Direction of the Incoming Ray
The ray of light is given by the parametric equations:
\[
\frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z - 3}{3}
\]
This implies the direction vector of the ray is \( \vec{d} = (1, 2, 3) \) and a point on the ray is \( P(1, 2, 3) \).
### Step 3: Find the Point of Intersection of the Ray and the Plane
Substituting the parametric equations of the ray into the plane equation:
\[
5(1 + t) + 3(2 + 2t) - 2(3 + 3t) - 4 = 0
\]
Expanding this gives:
\[
5 + 5t + 6 + 6t - 6 - 6t - 4 = 0
\]
Combining like terms:
\[
5t + 5 = 0 \implies t = -1
\]
Substituting \( t = -1 \) back into the parametric equations gives the intersection point:
\[
P' = (1 - 1, 2 - 2, 3 - 3) = (0, 0, 0)
\]
### Step 4: Find the Reflection Point
To find the reflection point \( P'' \), we can use the formula:
\[
P'' = P + 2(\text{projection of } \vec{d} \text{ onto } \vec{n})
\]
The normal vector \( \vec{n} = (5, 3, -2) \).
1. **Calculate the projection of \( \vec{d} \) onto \( \vec{n} \):**
\[
\text{proj}_{\vec{n}} \vec{d} = \frac{\vec{d} \cdot \vec{n}}{\|\vec{n}\|^2} \vec{n}
\]
\[
\vec{d} \cdot \vec{n} = 1 \cdot 5 + 2 \cdot 3 + 3 \cdot (-2) = 5 + 6 - 6 = 5
\]
\[
\|\vec{n}\|^2 = 5^2 + 3^2 + (-2)^2 = 25 + 9 + 4 = 38
\]
\[
\text{proj}_{\vec{n}} \vec{d} = \frac{5}{38}(5, 3, -2) = \left(\frac{25}{38}, \frac{15}{38}, -\frac{10}{38}\right)
\]
2. **Calculate the reflection point:**
\[
P'' = P + 2 \cdot \text{proj}_{\vec{n}} \vec{d}
\]
\[
P'' = (1, 2, 3) + 2\left(\frac{25}{38}, \frac{15}{38}, -\frac{10}{38}\right) = \left(1 + \frac{50}{38}, 2 + \frac{30}{38}, 3 - \frac{20}{38}\right)
\]
Simplifying gives:
\[
P'' = \left(\frac{88}{38}, \frac{76}{38}, \frac{94}{38}\right) = \left(\frac{44}{19}, \frac{38}{19}, \frac{47}{19}\right)
\]
### Step 5: Equation of the Reflected Ray
The direction of the reflected ray is given by the direction of the incoming ray reflected about the normal vector. The direction vector of the reflected ray can be calculated as:
\[
\vec{d_{reflected}} = \vec{d} - 2 \cdot \text{proj}_{\vec{n}} \vec{d}
\]
This will yield the direction of the reflected ray.
### Final Step: Write the Equation of the Reflected Ray
The equation of the reflected ray can be expressed as:
\[
\frac{x - x_0}{d_x} = \frac{y - y_0}{d_y} = \frac{z - z_0}{d_z}
\]
where \( (x_0, y_0, z_0) \) is the reflection point and \( (d_x, d_y, d_z) \) is the direction vector of the reflected ray.
