Two vectors \( (\hat{\imath}+\hat{\jmath}) \) and \( (3\hat{\imath}-\hat{\jmath}+4\hat{k}) \) represent the sides \( \overline{AB} \) and \( \overline{AC} \) vectors of a triangle \( ABC \) respectively, then the length of the median through the vertex \( A \) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 2\sqrt{2} \) units
If D is the mid-point of BC, the median through A is \( \overrightarrow{AD}=\frac{1}{2}\left(\overrightarrow{AB}+\overrightarrow{AC}\right) \).
\( \displaystyle \overrightarrow{AD}=\frac{1}{2}\left[(\hat{\imath}+\hat{\jmath})+(3\hat{\imath}-\hat{\jmath}+4\hat{k})\right] \)
\( \displaystyle =\frac{1}{2}(4\hat{\imath}+4\hat{k}) \)
\( \displaystyle =2\hat{\imath}+2\hat{k} \)
\( \displaystyle |\overrightarrow{AD}|=\sqrt{2^2+0^2+2^2} \)
\( \displaystyle =\sqrt{8}=2\sqrt{2}\ \text{units} \)
Why the other options are incorrect
Option (1) \( 4\sqrt{2} \) units — Incorrect. \( 4\sqrt{2} \) is \( |\overrightarrow{AB}+\overrightarrow{AC}| \); the factor \( \frac{1}{2} \) for the mid-point has been left out.
Option (3) \( \sqrt{2} \) units — Incorrect. \( \sqrt{2} \) is \( |\overrightarrow{AB}| \), the length of the side AB, not of the median.
Option (4) \( 3\sqrt{2} \) units — Incorrect. The median has length \( 2\sqrt{2} \), not \( 3\sqrt{2} \).