JEE Main 2022 (Online) 29th July Evening ShiftQuestion 1
Let $\vec a$ and $\vec b$ be two vectors such that $|\vec a+\vec b|^2=|\vec a|^2+2|\vec b|^2$, $\vec a\cdot\vec b=3$ and $|\vec a\times\vec b|^2=75$. Then $|\vec a|^2$ is equal to ________.
Vector Algebra
Answer$14$
Solution
$|\vec a+\vec b|^2=|\vec a|^2+2\vec a\cdot\vec b+|\vec b|^2=|\vec a|^2+2|\vec b|^2\Rightarrow 2\vec a\cdot\vec b=|\vec b|^2\Rightarrow|\vec b|^2=6$. $|\vec a\times\vec b|^2=|\vec a|^2|\vec b|^2-(\vec a\cdot\vec b)^2\Rightarrow75=6|\vec a|^2-9\Rightarrow|\vec a|^2=14$.