<p>The vectors \(\overrightarrow{AB} = 3\hat{i} + 4\hat{k}\) and \(\overrightarrow{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}\) are the sides of a triangle \(ABC\). The length of the median through \(A\) is</p>
Step-by-Step Solution
Key Concept: The median from vertex A goes to the midpoint M of BC. Use the property that M = (B+C)/2, so the median vector is $\overrightarrow{AM} = \frac{1}{2}(\overrightarrow{AB} + \overrightarrow{AC})$, then find its magnitude.
Step 1: Identify the median formula. The median through A to the midpoint M of BC is given by: $\overrightarrow{AM} = \frac{1}{2}(\overrightarrow{AB} + \overrightarrow{AC})$ Step 2: Substitute the given vectors: $\overrightarrow{AM} = \frac{1}{2}[(3\hat{i} + 4\hat{k}) + (5\hat{i} - 2\hat{j} + 4\hat{k})]$ $\overrightarrow{AM} = \frac{1}{2}[8\hat{i} - 2\hat{j} + 8\hat{k}]$ $\overrightarrow{AM} = 4\hat{i} - \hat{j} + 4\hat{k}$ Step 3: Calculate the magnitude: $|\overrightarrow{AM}| = \sqrt{4^2 + (-1)^2 + 4^2} = \sqrt{16 + 1 + 16} = \sqrt{33}$ ∴ Answer: C
Correct Answer: C