Question

The two vectors \(\hat{i}+\hat{j}+\hat{k}\) and \(\hat{i}+3\hat{j}+5\hat{k}\) represent the two sides \(\vec{AB}\) and \(\vec{AC}\) respectively of a ΔABC. The length of the median through A is

• A. \(\frac{\sqrt{14}}{2}\)
• B. 14
• C. 7
• D. \(\sqrt{14}\)

Answer 1

The Correct Option is D

The two vectors \(\hat{i} + \hat{j} + \hat{k}\) and \(\hat{i} + 3\hat{j} + 5\hat{k}\) represent the two sides \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) respectively of a \(\triangle ABC\). 

We need to find the length of the median through A.

Let D be the midpoint of BC. The median through A is AD. We can find \(\overrightarrow{AD}\) using the formula for the midpoint:

\(\overrightarrow{AD} = \frac{1}{2}(\overrightarrow{AB} + \overrightarrow{AC})\)

\(\overrightarrow{AD} = \frac{1}{2} ((\hat{i} + \hat{j} + \hat{k}) + (\hat{i} + 3\hat{j} + 5\hat{k}))\)

\(\overrightarrow{AD} = \frac{1}{2} (2\hat{i} + 4\hat{j} + 6\hat{k}) = \hat{i} + 2\hat{j} + 3\hat{k}\)

The length of the median is the magnitude of \(\overrightarrow{AD}\):

\(|\overrightarrow{AD}| = \sqrt{(1)^2 + (2)^2 + (3)^2} = \sqrt{1 + 4 + 9} = \sqrt{14}\)

Therefore, the length of the median through A is \(\sqrt{14}\).

Thus, the correct option is (D) \(\sqrt{14}\).

Answer 2

Let $ D $ be the midpoint of $ BC $. Then:

$$ \vec{AD} = \frac{1}{2} (\vec{AB} + \vec{AC}) = \frac{1}{2} \left( (\hat{i} + \hat{j} + \hat{k}) + (\hat{i} + 3\hat{j} + 5\hat{k}) \right). $$

Simplify:

$$ \vec{AD} = \frac{1}{2} (2\hat{i} + 4\hat{j} + 6\hat{k}) = \hat{i} + 2\hat{j} + 3\hat{k}. $$

The length of the median through $ A $ is:

$$ |\vec{AD}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14}. $$

Final Answer: The final answer is $ {\sqrt{14}} $.