The components of a particles velocity in the directions at right angles are `3 m//s,4 m//s "and "12m//s` respectively . The actual velocity of the particle , in `m//s` is

Answer

To find the actual velocity of the particle given its components in three perpendicular directions, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Components of Velocity**:
The components of the particle's velocity are given as:
- \( v_x = 3 \, \text{m/s} \) (along the x-direction)
- \( v_y = 4 \, \text{m/s} \) (along the y-direction)
- \( v_z = 12 \, \text{m/s} \) (along the z-direction)

2. **Write the Velocity Vector**:
The velocity vector \( \vec{v} \) can be expressed in vector form as:
\[ \vec{v} = v_x \hat{i} + v_y \hat{j} + v_z \hat{k} = 3 \hat{i} + 4 \hat{j} + 12 \hat{k} \, \text{m/s} \]

3. **Calculate the Magnitude of the Velocity Vector**:
The magnitude of the velocity vector \( |\vec{v}| \) is calculated using the formula:
\[ |\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2} \]
Substituting the values:
\[ |\vec{v}| = \sqrt{3^2 + 4^2 + 12^2} \]

4. **Compute the Squares**:
Calculate each square:
- \( 3^2 = 9 \)
- \( 4^2 = 16 \)
- \( 12^2 = 144 \)

5. **Sum the Squares**:
Add the squares together:
\[ 9 + 16 + 144 = 169 \]

6. **Take the Square Root**:
Now, take the square root of the sum:
\[ |\vec{v}| = \sqrt{169} = 13 \, \text{m/s} \]

### Final Answer:
The actual velocity of the particle is \( 13 \, \text{m/s} \).

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