JEE Main 2020 (Online) 2nd September Evening ShiftQuestion 1
Let the position vectors of points `$A$' and `$B$' be $\hat i+\hat j+\hat k$ and $2\hat i+\hat j+3\hat k$, respectively. A point `$P$' divides the line segment $AB$ internally in the ratio $\lambda:1$ ($\lambda>0$). If $O$ is the origin and $\overrightarrow{OB}\cdot\overrightarrow{OP}-3\left|\overrightarrow{OA}\times\overrightarrow{OP}\right|^2=6$, then $\lambda$ is equal to ________.
Vector Algebra
Answer$0.8$
Solution
$\vec{OP}=\frac{\vec A+\lambda\vec B}{\lambda+1}$ with $\vec A=(1,1,1),\vec B=(2,1,3)$. $\vec{OB}\cdot\vec{OP}=\frac{\lambda\cdot14+6}{\lambda+1}$ (since $\vec B\cdot\vec A=6$, $|\vec B|^2=14$). $\vec A\times\vec{OP}=\frac{\lambda\,\vec A\times\vec B}{\lambda+1}$, $\vec A\times\vec B=(2,-1,-1)$, so $|\cdot|^2=\frac{6\lambda^2}{(\lambda+1)^2}$. $(14\lambda+6)(\lambda+1)-18\lambda^2=6(\lambda+1)^2\Rightarrow10\lambda^2-8\lambda=0\Rightarrow\lambda=\frac45$ ($\lambda>0$).