Mathematics › Vectors › Product of 2 vectors
If $(\bar{a} \times \bar{b}) \times \bar{c}=-5 \bar{a}+4 \bar{b}$ and $\bar{a} \cdot \bar{b}=3$, then the…
If $(\bar{a} \times \bar{b}) \times \bar{c}=-5 \bar{a}+4 \bar{b}$ and $\bar{a} \cdot \bar{b}=3$, then the value of $\bar{a} \times(\bar{b} \times \bar{c})$ is
$3 \bar{b}-4 \bar{c}$ $4 \overline{\mathrm{a}}-3 \overline{\mathrm{b}}$ $4 \overline{\mathrm{b}}-3 \overline{\mathrm{c}}$ $3 \bar{a}-4 \bar{c}$ Solution
$\begin{aligned}
& (\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times \overline{\mathrm{c}}=(\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}) \overline{\mathrm{b}}-(\overline{\mathrm{b}} \cdot \overline{\mathrm{c}}) \overline{\mathrm{a}} \\
& \text { But, }(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \times \overline{\mathrm{c}}=-5 \overline{\mathrm{a}}+4 \overline{\mathrm{b}} \\
\therefore \quad & -5 \overline{\mathrm{a}}+4 \overline{\mathrm{b}}=(\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}) \overline{\mathrm{b}}-(\overline{\mathrm{b}} \cdot \overline{\mathrm{c}}) \overline{\mathrm{a}}
\end{aligned}$
Comparing, we get
$\begin{aligned}
\therefore \quad \overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=4 & \\
\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}}) & =(\overline{\mathrm{a} \cdot} \cdot \overline{\mathrm{c}}) \overline{\mathrm{b}}-(\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}) \overline{\mathrm{c}} \\
& =4 \overline{\mathrm{b}}-3 \overline{\mathrm{c}} \quad \ldots[\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=3 \text { (given) }]
\end{aligned}$
Asked in: MHT CET 2023 (10 May Shift 2)
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