To find the length of the median through point A in triangle ABC, we can follow these steps:
### Step 1: Identify the vectors
We are given the vectors:
- \(\vec{AB} = 2\hat{i} + 4\hat{j} + 4\hat{k}\)
- \(\vec{AC} = 2\hat{i} + 2\hat{j} + \hat{k}\)
### Step 2: Find the coordinates of points B and C
Assuming point A is at the origin (0, 0, 0):
- Point B can be represented as \(B(2, 4, 4)\)
- Point C can be represented as \(C(2, 2, 1)\)
### Step 3: Find the midpoint of segment BC
The midpoint \(M\) of segment \(BC\) can be calculated using the formula:
\[
\vec{M} = \frac{\vec{B} + \vec{C}}{2}
\]
Calculating the coordinates of midpoint \(M\):
\[
\vec{M} = \frac{(2, 4, 4) + (2, 2, 1)}{2} = \frac{(2+2, 4+2, 4+1)}{2} = \frac{(4, 6, 5)}{2} = (2, 3, 2.5)
\]
### Step 4: Find the vector from A to M
The vector \(\vec{AM}\) can be calculated as:
\[
\vec{AM} = \vec{M} - \vec{A}
\]
Since point A is at the origin:
\[
\vec{AM} = (2, 3, 2.5) - (0, 0, 0) = (2, 3, 2.5)
\]
### Step 5: Calculate the length of the median
The length of the median \(AM\) can be found using the formula for the magnitude of a vector:
\[
|\vec{AM}| = \sqrt{(2)^2 + (3)^2 + (2.5)^2}
\]
Calculating the squares:
\[
|\vec{AM}| = \sqrt{4 + 9 + 6.25} = \sqrt{19.25}
\]
Now, simplifying:
\[
|\vec{AM}| = \sqrt{19.25} \approx 4.39
\]
### Final Answer
The length of the median through A is approximately \(4.39\) units.
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