JEE Main 2021 (Online) 22nd July Evening ShiftQuestion 1
Let three vectors $\vec{a},\vec{b}$ and $\vec{c}$ be such that $\vec{a}\times\vec{b}=\vec{c}$, $\vec{b}\times\vec{c}=\vec{a}$ and $|\vec{a}|=2$. Then which one of the following is not true ?
Vector Algebra
Solution
$\vec c=\vec a\times\vec b$ is perpendicular to $\vec a,\vec b$, and $\vec a=\vec b\times\vec c$ is perpendicular to $\vec b,\vec c$, so $\vec a,\vec b,\vec c$ are mutually perpendicular. $|\vec c|=|\vec a||\vec b|$ and $|\vec a|=|\vec b||\vec c|=|\vec b|^2|\vec a|$ give $|\vec b|=1$, $|\vec c|=2$. (A) $[\vec a\,\vec b\,\vec c]=\vec a\cdot(\vec b\times\vec c)=|\vec a|^2=4$ and $[\vec c\,\vec a\,\vec b]$ is the same, so the sum is $8$: true. (B) $(\vec b+\vec c)\times(\vec b-\vec c)=-2\vec b\times\vec c=-2\vec a$, and $\vec a\times\vec a=0$: true. (C) $|3\vec a+\vec b-2\vec c|^2=9\cdot4+1+4\cdot4=53\ne51$: not true. (D) Projection of $\vec a$ on $\vec b\times\vec c=\vec a$ is $|\vec a|=2$: true. Option (C) is the false one.