Question

If \( \vec{AB} = 3\hat{i} + 5\hat{j} + 4\hat{k} \), \( \vec{AC} = 5\hat{i} - 5\hat{j} + 2\hat{k} \) represent the sides of triangle \( ABC \), then the length of the median through \( A \) is

• A. \( \sqrt{6} \) units
• B. \( 5 \) units
• C. \( \sqrt{5} \) units
• D. \( 6 \) units

Hint:

The median vector from a vertex is always half the sum of the other two side vectors.

Answer 1

The Correct Option is B

Step 1: Formula for median vector.
The vector of the median through vertex \( A \) is given by \[ \vec{AM} = \frac{1}{2}(\vec{AB} + \vec{AC}) \]

Step 2: Add the given vectors.
\[ \vec{AB} + \vec{AC} = (3+5)\hat{i} + (5-5)\hat{j} + (4+2)\hat{k} = 8\hat{i} + 0\hat{j} + 6\hat{k} \]

Step 3: Find the median vector.
\[ \vec{AM} = \frac{1}{2}(8\hat{i} + 6\hat{k}) = 4\hat{i} + 3\hat{k} \]

Step 4: Find the length of the median.
\[ |\vec{AM}| = \sqrt{4^2 + 3^2} = \sqrt{25} = 5 \]

Step 5: Conclusion.
The length of the median through \( A \) is \[ \boxed{5 \text{ units}} \]