The magnitude of the vector `6hat(i)+2hat(j)+3hat(k)` is :

Options

A. 5
B. 7
C. 12
D. 1

Answer

To find the magnitude of the vector \( \vec{v} = 6\hat{i} + 2\hat{j} + 3\hat{k} \), we can use the formula for the magnitude of a vector in three-dimensional space. The magnitude \( |\vec{v}| \) of a vector \( \vec{v} = a\hat{i} + b\hat{j} + c\hat{k} \) is given by:

\[ |\vec{v}| = \sqrt{a^2 + b^2 + c^2} \]

### Step-by-Step Solution:

1. **Identify the components of the vector**:
- Here, \( a = 6 \), \( b = 2 \), and \( c = 3 \).

2. **Square each component**:
- Calculate \( a^2 = 6^2 = 36 \)
- Calculate \( b^2 = 2^2 = 4 \)
- Calculate \( c^2 = 3^2 = 9 \)

3. **Sum the squares of the components**:
- \( a^2 + b^2 + c^2 = 36 + 4 + 9 = 49 \)

4. **Take the square root of the sum**:
- \( |\vec{v}| = \sqrt{49} = 7 \)

5. **Conclusion**:
- The magnitude of the vector \( 6\hat{i} + 2\hat{j} + 3\hat{k} \) is \( 7 \).

### Final Answer:
The magnitude of the vector \( 6\hat{i} + 2\hat{j} + 3\hat{k} \) is \( 7 \).