A particle has displacement of `12m` towards east and `5 m` towards north then `6 m` vertically upward. The sum of these displacements is-

Options

A. 12 m
B. 10.04 m
C. 14.31 m
D. 23 m

Explanation

To solve the problem of finding the resultant displacement of a particle that has moved in three perpendicular directions, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Displacements**:
The particle has three displacements:
- 12 m towards the east
- 5 m towards the north
- 6 m vertically upward

2. **Assign Unit Vectors**:
We can represent these displacements using unit vectors:
- East direction as \( \hat{i} \)
- North direction as \( \hat{j} \)
- Upward direction as \( \hat{k} \)

3. **Express the Displacement Vector**:
The total displacement vector \( \vec{r} \) can be expressed as:
\[ \vec{r} = 12 \hat{i} + 5 \hat{j} + 6 \hat{k} \]

4. **Calculate the Magnitude of the Displacement Vector**:
The magnitude \( r \) of the displacement vector can be calculated using the formula:
\[ r = \sqrt{(12)^2 + (5)^2 + (6)^2} \]
- Calculate \( 12^2 = 144 \)
- Calculate \( 5^2 = 25 \)
- Calculate \( 6^2 = 36 \)

5. **Sum the Squares**:
Now, add these values together:
\[ r = \sqrt{144 + 25 + 36} = \sqrt{205} \]

6. **Calculate the Square Root**:
Finally, find the square root of 205:
\[ r \approx 14.32 \text{ m} \]

### Final Answer:
The sum of the displacements is approximately \( 14.32 \text{ m} \).

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