To find the length of the median through point A in triangle ABC formed by the vectors \(\vec{AB} = 3\hat{i} + 4\hat{k}\) and \(\vec{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}\), we can follow these steps:
### Step 1: Find the coordinates of points A, B, and C
Assuming point A is at the origin (0, 0, 0):
- Point B can be represented as \(B(3, 0, 4)\) from vector \(\vec{AB}\).
- Point C can be represented as \(C(5, -2, 4)\) from vector \(\vec{AC}\).
### Step 2: Find the midpoint of segment BC
The midpoint \(M\) of segment \(BC\) can be calculated using the midpoint formula:
\[
M = \left(\frac{x_B + x_C}{2}, \frac{y_B + y_C}{2}, \frac{z_B + z_C}{2}\right)
\]
Substituting the coordinates of points B and C:
\[
M = \left(\frac{3 + 5}{2}, \frac{0 - 2}{2}, \frac{4 + 4}{2}\right) = \left(\frac{8}{2}, \frac{-2}{2}, \frac{8}{2}\right) = (4, -1, 4)
\]
### Step 3: Find the vector \(\vec{AM}\)
The vector \(\vec{AM}\) from point A to midpoint M is given by:
\[
\vec{AM} = M - A = (4, -1, 4) - (0, 0, 0) = 4\hat{i} - 1\hat{j} + 4\hat{k}
\]
### Step 4: Calculate the length of the median \(\vec{AM}\)
The length of the vector \(\vec{AM}\) can be calculated using the formula for the magnitude of a vector:
\[
|\vec{AM}| = \sqrt{(x^2 + y^2 + z^2)} = \sqrt{(4)^2 + (-1)^2 + (4)^2}
\]
Calculating this gives:
\[
|\vec{AM}| = \sqrt{16 + 1 + 16} = \sqrt{33}
\]
### Conclusion
The length of the median through point A is \(\sqrt{33}\).
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