In a triangle, \(\Delta\)ABC, the sides AB and AC are represented by vectors \(\hat{\imath} + \hat{\jmath} + \hat{k}\) and \(2\hat{\imath} - \hat{k}\) respectively. The length of median drawn from vertex A to BC is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\sqrt{\displaystyle\frac{5}{2}}\)
Let D be the midpoint of BC. The median vector from A is half the sum of the two sides from A:
\( \displaystyle \overrightarrow{AD} = \frac{\overrightarrow{AB} + \overrightarrow{AC}}{2} \)
\( \displaystyle = \frac{3\hat{\imath} + \hat{\jmath} + 0\hat{k}}{2} \)
Its length:
\( \displaystyle |\overrightarrow{AD}| = \frac{\sqrt{9 + 1 + 0}}{2} \)
\( \displaystyle = \frac{\sqrt{10}}{2} = \sqrt{\frac{10}{4}} = \sqrt{\frac{5}{2}} \)