A plane mirror coincides with a plane having equation x = 3. A particle is moving along a line with direction rations 3,4,5. If speed of the particle is `sqrt(2)`, the velocity of its image is

Explanation

To solve the problem, we need to determine the velocity of the image of a particle moving in the presence of a plane mirror located at \( x = 3 \).

### Step-by-Step Solution:

1. **Identify the Direction Ratios and Speed**:
The direction ratios of the particle's motion are given as \( 3, 4, 5 \) and its speed is \( \sqrt{2} \).

2. **Calculate the Velocity Components**:
The velocity vector \( \vec{v} \) of the particle can be expressed using the direction ratios and speed. The components of the velocity in the \( x, y, z \) directions are given by:
\[ \vec{v} = \left( \frac{3}{\sqrt{3^2 + 4^2 + 5^2}} \cdot \sqrt{2}, \frac{4}{\sqrt{3^2 + 4^2 + 5^2}} \cdot \sqrt{2}, \frac{5}{\sqrt{3^2 + 4^2 + 5^2}} \cdot \sqrt{2} \right) \]
First, we calculate the magnitude of the direction ratios:
\[ \sqrt{3^2 + 4^2 + 5^2} = \sqrt{9 + 16 + 25} = \sqrt{50} = 5\sqrt{2} \]
Thus, the components of the velocity become:
\[ \vec{v} = \left( \frac{3}{5\sqrt{2}} \cdot \sqrt{2}, \frac{4}{5\sqrt{2}} \cdot \sqrt{2}, \frac{5}{5\sqrt{2}} \cdot \sqrt{2} \right) = \left( \frac{3}{5}, \frac{4}{5}, 1 \right) \]

3. **Determine the Effect of the Mirror**:
The mirror is located at \( x = 3 \). When a particle's image is formed in a plane mirror, the component of the velocity that is perpendicular to the mirror (the \( x \)-component) will reverse its direction, while the components parallel to the mirror (the \( y \) and \( z \)-components) will remain unchanged.

4. **Calculate the Image Velocity**:
The original velocity vector is:
\[ \vec{v} = \left( \frac{3}{5}, \frac{4}{5}, 1 \right) \]
The image velocity \( \vec{v}_{\text{image}} \) will have the \( x \)-component inverted:
\[ \vec{v}_{\text{image}} = \left( -\frac{3}{5}, \frac{4}{5}, 1 \right) \]

5. **Final Result**:
The velocity of the image of the particle is:
\[ \vec{v}_{\text{image}} = \left( -\frac{3}{5}, \frac{4}{5}, 1 \right) \]