If A(3,1,-1), B(\frac53, \frac73, \frac13), C(2,2,1) and D(\frac103, \frac23, \frac-13) are the vertices of a quadrilateral A B C D , then its area is

3D Geometry PYQ — JEE Main 2024 · 6 Apr · Shift 1 · Q35 with Solution

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Correct answer: A

Given $A(3,1,-1),\quad B\left(\frac53,\frac73,\frac13\right),\quad C(2,2,1),\quad D\left(\frac{10}3,\frac23,-\frac13\right).$

Let us compute the consecutive side vectors:

$\overrightarrow{AB}=B-A=\left(\frac53-3,\frac73-1,\frac13-(-1)\right)=\left(-\frac43,\frac43,\frac43\right).$

$\overrightarrow{BC}=C-B=\left(2-\frac53,2-\frac73,1-\frac13\right)=\left(\frac13,-\frac13,\frac23\right).$

$\overrightarrow{CD}=D-C=\left(\frac{10}3-2,\frac23-2,-\frac13-1\right)=\left(\frac43,-\frac43,-\frac43\right)=-\overrightarrow{AB}.$

$\overrightarrow{DA}=A-D=\left(3-\frac{10}3,1-\frac23,-1-\left(-\frac13\right)\right)=\left(-\frac13,\frac13,-\frac23\right)=-\overrightarrow{BC}.$

So opposite sides are equal and parallel. Hence $ABCD$ is a parallelogram.

Area of parallelogram $ABCD$ is $\text{Area}=\left|\overrightarrow{AB}\times\overrightarrow{BC}\right|.$

Now,

\qquad \overrightarrow{BC}=\left(\frac13,-\frac13,\frac23\right).$$ Compute the cross product: $$\overrightarrow{AB}\times\overrightarrow{BC} =\begin{vmatrix} \hat i & \hat j & \hat k \\ -\frac43 & \frac43 & \frac43 \\ \frac13 & -\frac13 & \frac23 \end{vmatrix}.$$ $$=\hat i\left(\frac43\cdot\frac23-\frac43\cdot\left(-\frac13\right)\right) -\hat j\left(-\frac43\cdot\frac23-\frac43\cdot\frac13\right) +\hat k\left(-\frac43\cdot\left(-\frac13\right)-\frac43\cdot\frac13\right).$$ $$=\hat i\left(\frac89+\frac49\right)-\hat j\left(-\frac89-\frac49\right)+\hat k\left(\frac49-\frac49\right).$$ $$=\left(\frac43,\frac43,0\right).$$ Therefore, $$\text{Area}=\sqrt{\left(\frac43\right)^2+\left(\frac43\right)^2} =\frac43\sqrt{2} =\frac{4\sqrt{2}}{3}.$$ --- 3. **Match with options** Thus the area of quadrilateral $ABCD$ is $$\boxed{\frac{4\sqrt{2}}{3}}.$$ So the correct option is **A**. --- 4. **Comparison with stored answer** Stored correct answer: **A** Our derived answer: **A** Hence, they agree.