JEE Advanced 2019 Paper 1Question 1
Three lines are given by $$\vec r=\lambda\hat i,\ \lambda\in\mathbb{R}$$ $$\vec r=\mu(\hat i+\hat j),\ \mu\in\mathbb{R}\ \text{and}$$ $$\vec r=\nu(\hat i+\hat j+\hat k),\ \nu\in\mathbb{R}.$$ Let the lines cut the plane $x+y+z=1$ at the points $A$, $B$ and $C$ respectively. If the area of the triangle $ABC$ is $\Delta$ then the value of $(6\Delta)^2$ equals ____
Vector Algebra
Answer$0.75$
Solution
$A=(1,0,0)$, $B=(\frac12,\frac12,0)$, $C=(\frac13,\frac13,\frac13)$. $\vec{AB}\times\vec{AC}=\frac16(1,1,1)$, so $\Delta=\frac12\cdot\frac{\sqrt3}{6}=\frac{\sqrt3}{12}$ and $(6\Delta)^2=\frac34$.