If A(3,1,-1) , B ((5)/(3),(7)/(3),(1)/(3) ) , C(2,2,1) and D ((10)/(3),(2)/(3),(-1)/(3) ) are the vertices of a quadrilateral ABCD , then its area is

3D Geometry: If A(3,1,-1) , B ((5)/(3),(7)/(3),(1)/(3) ) , C(2,2,1) and D… | JEE Main 2024

JEE Main 2024 (Online) 6th April Morning ShiftQuestion 1

If $A(3,1,-1)$, $B\left(\frac{5}{3},\frac{7}{3},\frac{1}{3}\right)$, $C(2,2,1)$ and $D\left(\frac{10}{3},\frac{2}{3},\frac{-1}{3}\right)$ are the vertices of a quadrilateral $ABCD$, then its area is

3D Geometry

$2\sqrt{2}$$\frac{5\sqrt{2}}{3}$$\frac{2\sqrt{2}}{3}$$\frac{4\sqrt{2}}{3}$✓ Correct answer

Solution

Points are coplanar; area $=\frac12|\vec{AC}\times\vec{BD}|$ with $\vec{AC}=(-1,1,2)$, $\vec{BD}=(\frac53,-\frac53,-\frac23)$, cross $=(\frac83,\frac83,0)$. Area $=\frac12\cdot\frac{8\sqrt2}3=\frac{4\sqrt2}3$.

3D Geometry - JEE Mathematics question (JEE Main 2024 (Online) 6th April Morning Shift)

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