For non-zero vectors $\bar{a}, \bar{b}, \bar{c},|(\bar{a} \times \bar{b}) \cdot…
For non-zero vectors $\bar{a}, \bar{b}, \bar{c},|(\bar{a} \times \bar{b}) \cdot \bar{c}|=|\bar{a}||\bar{b}||\bar{c}|$ holds and only if
- $\bar{b} \cdot \bar{c}=0, \bar{c} \cdot \bar{a}=0$
- $\bar{c} \cdot \bar{a}=0, \bar{a} \cdot \bar{b}=0$
- $\bar{a} \cdot \bar{b}=\bar{b} \cdot \bar{c}=\bar{c} \cdot \bar{a}=0$
- $\bar{a} \times \bar{b}=0, \bar{b} \times \bar{c}=0$
Solution
$\begin{aligned} & |(\vec{a} \times \vec{b}) \cdot \vec{c}| \\ & =|\vec{a}||\vec{b}||\vec{c}| \\ & \Rightarrow|\vec{a} \times \vec{b}||\vec{c}| \cos \theta=|\vec{a}||\vec{b}||\vec{c}| \\ & \Rightarrow|\vec{a}||\vec{b}| \sin \phi|\vec{c}| \cos \theta=|\vec{a}||\vec{b}||\vec{c}| \\ & \Rightarrow \sin \phi=1 \text { and } \cos \theta=1 \\ & \phi=90^{\circ} \text { and } \theta=0^{\circ} \\ & \text { where } \phi \text { is the angle between } \vec{a} \text { and } \vec{b} \text { and } \theta \text { is the angle between } \vec{c} \text { and } \\ & \text { normal to the plane containing } \vec{a} \text { and } \vec{b} \\ & \Rightarrow \vec{a} \perp \vec{b}, \vec{b} \perp \vec{c} \text {, and } \vec{c} \perp \vec{a} \\ & \Rightarrow \vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{c}=\vec{c} \cdot \vec{a}=0\end{aligned}$
Asked in: MHT CET 2022 (08 Aug Shift 2)
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