Question

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

• A. \(\sqrt{72}\)
• B. \(\sqrt{33}\)
• C. \(\sqrt{288}\)
• D. \(\sqrt{18}\)

Hint:

For a triangle with sides \(\vec{AB}\) and \(\vec{AC}\), the median through vertex \(A\) is: \[ \vec{AM}=\frac{\vec{AB}+\vec{AC}}{2} \] Always find the magnitude of the median vector for its length.

Answer 1

The Correct Option is D

Step 1: The vector representing the median through vertex \(A\) is given by: \[ \vec{AM}=\frac{1}{2}\left(\vec{AB}+\vec{AC}\right) \]
Step 2: Write the given vectors in component form: \[ \vec{AB}=(-3,\,0,\,4), \qquad \vec{AC}=(5,\,-2,\,4) \]
Step 3: Add the vectors: \[ \vec{AB}+\vec{AC}=(2,\,-2,\,8) \]
Step 4: Find the median vector: \[ \vec{AM}=\frac{1}{2}(2,\,-2,\,8)=(1,\,-1,\,4) \]
Step 5: Length of the median: \[ |\vec{AM}|=\sqrt{1^2+(-1)^2+4^2} =\sqrt{1+1+16} =\sqrt{18} \]