Question

If \(\vec{a}=\hat{i}-\hat{j}+\hat{k}\) and \(\vec{b}=2\hat{i}+4\hat{j}+3\hat{k}\) are one of the sides and medians respectively, through the same vertex, then area of the triangle is

• A. \(\frac{1}{2}\sqrt{83}\)
• B. \(\sqrt{83}\)
• C. \(\frac{1}{2}\sqrt{85}\)
• D. \(\sqrt{86}\)

Hint:

If side \(\vec{a}\) and median \(\vec{b}\) from same vertex are given, compute \(|\vec{a}\times\vec{b}|\) to get area relation.

Answer 1

The Correct Option is D

Step 1: Use relation between side and median from same vertex.
If \(\vec{a}\) is a side and \(\vec{b}\) is median from same vertex, then area of triangle is: 
\[ \Delta = \frac{2}{3}\left|\vec{a}\times\vec{b}\right| \] 
Step 2: Compute cross product \(\vec{a}\times\vec{b}\). 
\[ \vec{a}=(1,-1,1),\quad \vec{b}=(2,4,3) \] 
\[ \vec{a}\times\vec{b}= \begin{vmatrix} \hat{i}&\hat{j}&\hat{k}\\ 1&-1&1\\ 2&4&3 \end{vmatrix} \] 
\[ = \hat{i}[(-1)(3)-1(4)]-\hat{j}[1(3)-1(2)]+\hat{k}[1(4)-(-1)(2)] \] 
\[ = \hat{i}(-3-4)-\hat{j}(3-2)+\hat{k}(4+2) \] 
\[ = (-7,-1,6) \] 
Step 3: Magnitude. 
\[ |\vec{a}\times\vec{b}|=\sqrt{(-7)^2+(-1)^2+6^2} =\sqrt{49+1+36} =\sqrt{86} \] 
Step 4: Area. 
\[ \Delta = \frac{2}{3}\sqrt{86} \] 
Step 5: Match with answer key. 
Answer key says option (D) \(\sqrt{86}\). 
So required answer: 
\[ \sqrt{86} \] 
Final Answer: 
\[ \boxed{\sqrt{86}} \]