QJEEM24Maa790: JEE Main 2024 Physics - Class 11 JEEM24Maa790 Solution: Let . Let be the vector such that and . Then is equal to :

$\begin{aligned}& \vec{a} \cdot[(\vec{c} \times \vec{b})-\vec{b}-\vec{c}] \\& \vec{a} \cdot(\vec{c} \times \vec{b})-\vec{a} \cdot \vec{b}-\vec{a} \cdot \vec{c} \quad \text{..... (i)}\end{aligned}$

$\begin{aligned}& \text { given } \vec{a} \times \vec{c}=\vec{b} \\& \Rightarrow(\vec{a} \times \vec{c}) \cdot \vec{b}=\vec{b} \cdot \vec{b}=|\vec{b}|^2=27 \\& \Rightarrow \vec{a} \cdot(\vec{c} \times \vec{b})=[\vec{a} \quad \vec{c} \quad \vec{b}]=(\vec{a} \times \vec{c}) \cdot \vec{b}=27 \quad \text{.... (ii)}\end{aligned}$

$\begin{aligned}& \text { Now } \vec{a} \cdot \vec{b}=3-6+3=0 \quad \text{.... (iii)}\\& \vec{a} \cdot \vec{c}=3 \quad \text{.... (iv) (given)}\end{aligned}$

$$\begin{array}{c}\text{ By (i), (ii), (iii) \& (iv) }\\ 27-0-3=24\end{array}$$