If the vectors `vec(A) B = 3 hat(i)+4 hat (k) and vec(AC) = 5 hat(i)-2 hat(j)+4hat(k)` are the sides of a triangle ABC, then the length of the median through A is

Options

A. `sqrt(18)`
B. `sqrt(72)`
C. `sqrt(33)`
D. `sqrt(45)`

Explanation

To find the length of the median through point A in triangle ABC, we will follow these steps:

### Step 1: Understand the Given Vectors
We are given two vectors:
- \( \vec{AB} = 3\hat{i} + 4\hat{k} \)
- \( \vec{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k} \)

These vectors represent the sides of triangle ABC.

### Step 2: Find the Midpoint of Side BC
The median from vertex A to side BC is represented by the vector \( \vec{AD} \), where D is the midpoint of side BC. To find \( \vec{AD} \), we first need to find the coordinates of point D.

The midpoint D can be found using the formula:
\[ \vec{D} = \frac{\vec{B} + \vec{C}}{2} \]
However, we can directly find \( \vec{AD} \) using the vectors \( \vec{AB} \) and \( \vec{AC} \).

### Step 3: Calculate the Median Vector \( \vec{AD} \)
The median vector \( \vec{AD} \) is given by:
\[ \vec{AD} = \frac{\vec{AB} + \vec{AC}}{2} \]

Substituting the values of \( \vec{AB} \) and \( \vec{AC} \):
\[ \vec{AD} = \frac{(3\hat{i} + 4\hat{k}) + (5\hat{i} - 2\hat{j} + 4\hat{k})}{2} \]

### Step 4: Simplify the Expression
Now, we simplify the expression:
\[ \vec{AD} = \frac{(3 + 5)\hat{i} + (-2)\hat{j} + (4 + 4)\hat{k}}{2} \]
\[ \vec{AD} = \frac{8\hat{i} - 2\hat{j} + 8\hat{k}}{2} \]
\[ \vec{AD} = 4\hat{i} - \hat{j} + 4\hat{k} \]

### Step 5: Calculate the Length of the Median
To find the length of the median \( AD \), we calculate the magnitude of \( \vec{AD} \):
\[ |\vec{AD}| = \sqrt{(4)^2 + (-1)^2 + (4)^2} \]
\[ |\vec{AD}| = \sqrt{16 + 1 + 16} \]
\[ |\vec{AD}| = \sqrt{33} \]

### Final Answer
Thus, the length of the median through A is:
\[ \sqrt{33} \]