The vector `vec(AB)=3hati+4hatk and vec(AC)=5hati-2hatj+4hatk` are sides of a triangle ABC. The length of the median through A is (A) `sqrt(18)` (B) `sqrt(72)` (C) `sqrt(33)` (D) `sqrt(288)`

Options

A. `sqrt(14)`
B. `sqrt(18)`
C. `sqrt(29)`
D. `5`

Explanation

To find the length of the median through point A in triangle ABC with given vectors \( \vec{AB} \) and \( \vec{AC} \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Vectors:**
- \( \vec{AB} = 3\hat{i} + 0\hat{j} + 4\hat{k} \)
- \( \vec{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k} \)

2. **Use the Median Formula:**
The median from vertex A to the midpoint D of side BC can be calculated using the formula:
\[ \vec{AD} = \frac{1}{2}(\vec{AB} + \vec{AC}) \]
This means we first need to calculate \( \vec{AB} + \vec{AC} \).

3. **Calculate \( \vec{AB} + \vec{AC} \):**
\[ \vec{AB} + \vec{AC} = (3\hat{i} + 0\hat{j} + 4\hat{k}) + (5\hat{i} - 2\hat{j} + 4\hat{k}) \]
Combine the components:
- \( i \) components: \( 3 + 5 = 8 \)
- \( j \) components: \( 0 - 2 = -2 \)
- \( k \) components: \( 4 + 4 = 8 \)

Thus,
\[ \vec{AB} + \vec{AC} = 8\hat{i} - 2\hat{j} + 8\hat{k} \]

4. **Calculate \( \vec{AD} \):**
Now, substitute into the median formula:
\[ \vec{AD} = \frac{1}{2}(8\hat{i} - 2\hat{j} + 8\hat{k}) = 4\hat{i} - \hat{j} + 4\hat{k} \]

5. **Find the Magnitude of \( \vec{AD} \):**
The length of the median \( AD \) is given by the magnitude:
\[ |\vec{AD}| = \sqrt{(4)^2 + (-1)^2 + (4)^2} \]
Calculate each term:
- \( (4)^2 = 16 \)
- \( (-1)^2 = 1 \)
- \( (4)^2 = 16 \)

Therefore,
\[ |\vec{AD}| = \sqrt{16 + 1 + 16} = \sqrt{33} \]

6. **Conclusion:**
The length of the median through A is \( \sqrt{33} \). Thus, the correct answer is option (C) \( \sqrt{33} \).